Polynomials are algebraic expressions that consist of terms combined by addition, subtraction, and multiplication. These terms are made up of variables (like $x$, $y$) raised to non-negative integer powers and constants. Factoring is the process of breaking down a polynomial into simpler components called factors, which, when multiplied together, give the original polynomial.
1. Definition of a Polynomial
A polynomial is an expression of the form:
$ a_nx^n + a_{n-1}x^{n-1} + \dots + a_1x + a_0 $
Where:
- $a_n, a_{n-1}, \dots, a_0$ are constants called coefficients
- $x$ is the variable
- $n$ is a non-negative integer representing the degree of the polynomial.
Example 1:
$ 3x^2 + 2x + 7 $ is a polynomial of degree 2, with coefficients 3, 2, and 7.
2. Types of Polynomials
- Monomial: A polynomial with only one term.
Example: $ 5x^3 $ - Binomial: A polynomial with two terms.
Example: $ x^2 + 3x $ - Trinomial: A polynomial with three terms.
Example: $ 4x^2 + 5x + 6 $ - Degree of a Polynomial: The highest power of the variable in the polynomial.
Example: The degree of $ 6x^5 + 3x^3 – 2x $ is 5.
3. Operations on Polynomials
Addition and Subtraction of Polynomials:
To add or subtract polynomials, combine like terms, which are terms with the same powers of the variables.
Example 2:
Add $ 4x^2 + 3x + 5 $ and $ 2x^2 – 4x + 1 $.
$ (4x^2 + 3x + 5) + (2x^2 – 4x + 1) $
$ = 4x^2 + 2x^2 + 3x – 4x + 5 + 1 $
$ = 6x^2 – x + 6 $.
Example 3:
Subtract $ 5x^2 + 4x – 7 $ from $ 3x^2 – 2x + 9 $.
$ (3x^2 – 2x + 9) – (5x^2 + 4x – 7) $
$ = 3x^2 – 5x^2 – 2x – 4x + 9 + 7 $
$ = -2x^2 – 6x + 16 $.
Multiplication of Polynomials:
Multiply each term in the first polynomial by each term in the second polynomial and then combine like terms.
Example 4:
Multiply $ (2x + 3) $ and $ (x^2 – x + 4) $.
$ (2x + 3)(x^2 – x + 4) $
$ = 2x(x^2 – x + 4) + 3(x^2 – x + 4) $
$ = 2x^3 – 2x^2 + 8x + 3x^2 – 3x + 12 $
$ = 2x^3 + x^2 + 5x + 12 $.
4. Factoring Polynomials
Factoring is the reverse of expanding polynomials. We break down a polynomial into factors that, when multiplied together, give back the original polynomial.
4.1 Factoring by Common Factor
Look for the greatest common factor (GCF) in all the terms and factor it out.
Example 5:
Factor $ 6x^3 + 9x^2 + 3x $.
The GCF is $ 3x $.
$ 6x^3 + 9x^2 + 3x = 3x(2x^2 + 3x + 1) $.
4.2 Factoring Trinomials
For trinomials of the form $ ax^2 + bx + c $, look for two numbers that multiply to give $ ac $ and add to give $ b $.
Example 6:
Factor $ x^2 + 5x + 6 $.
Find two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.
$ x^2 + 5x + 6 = (x + 2)(x + 3) $.
4.3 Difference of Squares
For expressions of the form $ a^2 – b^2 $, we can factor them as:
$ a^2 – b^2 = (a + b)(a – b) $.
Example 7:
Factor $ 9x^2 – 25 $.
$ 9x^2 – 25 = (3x + 5)(3x – 5) $.
4.4 Perfect Square Trinomials
Perfect square trinomials take the form:
$ a^2 + 2ab + b^2 = (a + b)^2 $
$ a^2 – 2ab + b^2 = (a – b)^2 $.
Example 8:
Factor $ x^2 + 6x + 9 $.
$ x^2 + 6x + 9 = (x + 3)^2 $.
5. Special Cases of Factoring
5.1 Cubic Polynomials
Sum of Cubes:
For expressions of the form $ a^3 + b^3 $, we can factor them as:
$ a^3 + b^3 = (a + b)(a^2 – ab + b^2) $.
Example 9:
Factor $ 8x^3 + 27 $.
$ 8x^3 + 27 = (2x + 3)((2x)^2 – 2x(3) + 3^2) $
$ = (2x + 3)(4x^2 – 6x + 9) $.
5.2 Factoring Polynomials with Four Terms
When a polynomial has four terms, we can often group the terms to factor them.
Example 10:
Factor $ 2x^3 + 6x^2 + 4x + 12 $.
Group the terms:
$ (2x^3 + 6x^2) + (4x + 12) $
Factor out the GCF:
$ 2x^2(x + 3) + 4(x + 3) $
Now factor out the common binomial:
$ (2x^2 + 4)(x + 3) $.
6. Application of Factoring in Solving Polynomial Equations
Factoring is often used to solve polynomial equations. After factoring, set each factor equal to zero and solve for the variable.
Example 11:
Solve $ x^2 – 5x + 6 = 0 $.
Factor the quadratic:
$ x^2 – 5x + 6 = (x – 2)(x – 3) $.
Set each factor equal to zero:
$ x – 2 = 0 $ or $ x – 3 = 0 $.
Solve:
$ x = 2 $ or $ x = 3 $.
7. Examples 12 to 100
Example 12:
Factor $ 4x^2 – 16 $.
Solution:
This is a difference of squares:
$ 4x^2 – 16 = (2x + 4)(2x – 4) $.
Example 13:
Factor $ x^2 – 9 $.
Solution:
$ x^2 – 9 = (x + 3)(x – 3) $.
Example 14:
Factor $ x^2 + 4x + 4 $.
Solution:
$ x^2 + 4x + 4 = (x + 2)^2 $.
Example 15:
Factor $ 3x^2 + 15x + 18 $.
Solution:
First, factor out the GCF:
$ 3(x^2 + 5x + 6) $.
Then factor the quadratic:
$ 3(x + 2)(x + 3) $.
Example 16:
Factor $ x^2 – 10x + 25 $.
Solution:
Recognize this as a perfect square trinomial:
$ x^2 – 10x + 25 = (x – 5)^2 $.
Example 17:
Factor $ 2x^2 – 8x $.
Solution:
Factor out the GCF:
$ 2x(x – 4) $.
Example 18:
Factor $ 4x^2 + 12x + 9 $.
Solution:
This is a perfect square trinomial:
$ 4x^2 + 12x + 9 = (2x + 3)^2 $.
Example 19:
Factor $ x^2 + 11x + 24 $.
Solution:
Find two numbers that multiply to 24 and add to 11. These numbers are 3 and 8.
$ x^2 + 11x + 24 = (x + 3)(x + 8) $.
Example 20:
Factor $ x^2 – 16x + 64 $.
Solution:
This is a perfect square trinomial:
$ x^2 – 16x + 64 = (x – 8)^2 $.
Example 21:
Factor $ 9x^2 – 1 $.
Solution:
This is a difference of squares:
$ 9x^2 – 1 = (3x + 1)(3x – 1) $.
Example 22:
Factor $ 6x^2 + 11x – 10 $.
Solution:
Find two numbers that multiply to $ 6(-10) = -60 $ and add to 11. These numbers are 15 and -4.
Rewrite the middle term and factor by grouping:
$ 6x^2 + 15x – 4x – 10 $
$ = 3x(2x + 5) – 2(2x + 5) $
$ = (3x – 2)(2x + 5) $.
Example 23:
Factor $ x^3 – 8 $.
Solution:
This is a difference of cubes:
$ x^3 – 8 = (x – 2)(x^2 + 2x + 4) $.
Example 24:
Factor $ 4x^2 + 20x + 25 $.
Solution:
Recognize this as a perfect square trinomial:
$ 4x^2 + 20x + 25 = (2x + 5)^2 $.
Example 25:
Factor $ x^4 – 16 $.
Solution:
This is a difference of squares:
$ x^4 – 16 = (x^2 + 4)(x^2 – 4) $.
Then, factor $ x^2 – 4 $:
$ (x^2 + 4)(x + 2)(x – 2) $.
Example 26:
Factor $ 3x^2 + 18x + 27 $.
Solution:
Factor out the GCF first:
$ 3(x^2 + 6x + 9) $.
Then factor the trinomial:
$ 3(x + 3)^2 $.
Example 27:
Factor $ x^2 + 7x + 10 $.
Solution:
Find two numbers that multiply to 10 and add to 7. These numbers are 5 and 2.
$ x^2 + 7x + 10 = (x + 5)(x + 2) $.
Example 28:
Factor $ 8x^2 + 14x – 15 $.
Solution:
Find two numbers that multiply to $ 8(-15) = -120 $ and add to 14. These numbers are 20 and -6.
Rewrite the middle term and factor by grouping:
$ 8x^2 + 20x – 6x – 15 $
$ = 4x(2x + 5) – 3(2x + 5) $
$ = (4x – 3)(2x + 5) $.
Example 29:
Factor $ 4x^2 – 25 $.
Solution:
This is a difference of squares:
$ 4x^2 – 25 = (2x + 5)(2x – 5) $.
Example 30:
Factor $ x^3 + 27 $.
Solution:
This is a sum of cubes:
$ x^3 + 27 = (x + 3)(x^2 – 3x + 9) $.
Example 31:
Factor $ x^2 + 10x + 16 $.
Solution:
Find two numbers that multiply to 16 and add to 10. These numbers are 8 and 2.
$ x^2 + 10x + 16 = (x + 8)(x + 2) $.
Example 32:
Factor $ 5x^2 – 20x $.
Solution:
Factor out the GCF:
$ 5x(x – 4) $.
Example 33:
Factor $ x^2 – 14x + 49 $.
Solution:
Recognize this as a perfect square trinomial:
$ x^2 – 14x + 49 = (x – 7)^2 $.
Example 34:
Factor $ 2x^2 – 3x – 2 $.
Solution:
Find two numbers that multiply to $ 2(-2) = -4 $ and add to -3. These numbers are -4 and 1.
Rewrite the middle term and factor by grouping:
$ 2x^2 – 4x + x – 2 $
$ = 2x(x – 2) + 1(x – 2) $
$ = (2x + 1)(x – 2) $.
Example 35:
Factor $ 9x^2 – 4 $.
Solution:
This is a difference of squares:
$ 9x^2 – 4 = (3x + 2)(3x – 2) $.
Example 36:
Factor $ x^3 – 125 $.
Solution:
This is a difference of cubes:
$ x^3 – 125 = (x – 5)(x^2 + 5x + 25) $.
Example 37:
Factor $ 6x^2 + 13x + 6 $.
Solution:
Find two numbers that multiply to $ 6(6) = 36 $ and add to 13. These numbers are 9 and 4.
Rewrite the middle term and factor by grouping:
$ 6x^2 + 9x + 4x + 6 $
$ = 3x(2x + 3) + 2(2x + 3) $
$ = (3x + 2)(2x + 3) $.
Example 38:
Factor $ x^2 – 49 $.
Solution:
This is a difference of squares:
$ x^2 – 49 = (x + 7)(x – 7) $.
Example 39:
Factor $ x^4 – 16x^2 $.
Solution:
Factor out the common factor:
$ x^2(x^2 – 16) $.
Then factor $ x^2 – 16 $:
$ x^2(x + 4)(x – 4) $.
Example 40:
Factor $ 16x^2 – 9 $.
Solution:
This is a difference of squares:
$ 16x^2 – 9 = (4x + 3)(4x – 3) $.
Example 41:
Factor $ x^2 + 6x – 16 $.
Solution:
Find two numbers that multiply to -16 and add to 6. These numbers are 8 and -2.
$ x^2 + 6x – 16 = (x + 8)(x – 2) $.
Example 42:
Factor $ 5x^2 + 15x $.
Solution:
Factor out the GCF:
$ 5x(x + 3) $.
Example 43:
Factor $ x^2 – 25x + 100 $.
Solution:
This is a perfect square trinomial:
$ x^2 – 25x + 100 = (x – 10)^2 $.
Example 44:
Factor $ x^3 + 64 $.
Solution:
This is a sum of cubes:
$ x^3 + 64 = (x + 4)(x^2 – 4x + 16) $.
Example 45:
Factor $ 4x^2 – 12x + 9 $.
Solution:
Recognize this as a perfect square trinomial:
$ 4x^2 – 12x + 9 = (2x – 3)^2 $.
Example 46:
Factor $ 5x^2 – 30x + 25 $.
Solution:
First, factor out the GCF:
$ 5(x^2 – 6x + 5) $.
Then factor the trinomial:
$ 5(x – 5)(x – 1) $.
Example 47:
Factor $ 2x^2 – 50 $.
Solution:
Factor out the GCF:
$ 2(x^2 – 25) $.
Then recognize $ x^2 – 25 $ as a difference of squares:
$ 2(x + 5)(x – 5) $.
Example 48:
Factor $ x^2 + 12x + 36 $.
Solution:
This is a perfect square trinomial:
$ x^2 + 12x + 36 = (x + 6)^2 $.
Example 49:
Factor $ 3x^3 + 9x^2 + 6x $.
Solution:
Factor out the GCF:
$ 3x(x^2 + 3x + 2) $.
Now factor the quadratic:
$ 3x(x + 2)(x + 1) $.
Example 50:
Factor $ x^2 – 1 $.
Solution:
This is a difference of squares:
$ x^2 – 1 = (x + 1)(x – 1) $.
Example 51:
Factor $ x^2 – 4x – 12 $.
Solution:
Find two numbers that multiply to -12 and add to -4. These numbers are -6 and 2.
$ x^2 – 4x – 12 = (x – 6)(x + 2) $.
Example 52:
Factor $ x^3 + 125 $.
Solution:
This is a sum of cubes:
$ x^3 + 125 = (x + 5)(x^2 – 5x + 25) $.
Example 53:
Factor $ 9x^2 – 16 $.
Solution:
This is a difference of squares:
$ 9x^2 – 16 = (3x + 4)(3x – 4) $.
Example 54:
Factor $ 2x^2 + 9x – 5 $.
Solution:
Find two numbers that multiply to $ 2(-5) = -10 $ and add to 9. These numbers are 10 and -1.
Rewrite the middle term and factor by grouping:
$ 2x^2 + 10x – x – 5 $
$ = 2x(x + 5) – 1(x + 5) $
$ = (2x – 1)(x + 5) $.
Example 55:
Factor $ 16x^2 + 40x + 25 $.
Solution:
This is a perfect square trinomial:
$ 16x^2 + 40x + 25 = (4x + 5)^2 $.
Example 56:
Factor $ x^2 – 100 $.
Solution:
This is a difference of squares:
$ x^2 – 100 = (x + 10)(x – 10) $.
Example 57:
Factor $ 3x^2 – 6x + 9 $.
Solution:
Factor out the GCF:
$ 3(x^2 – 2x + 3) $.
Since $ x^2 – 2x + 3 $ cannot be factored further, the final answer is:
$ 3(x^2 – 2x + 3) $.
Example 58:
Factor $ 2x^2 + 5x – 3 $.
Solution:
Find two numbers that multiply to $ 2(-3) = -6 $ and add to 5. These numbers are 6 and -1.
Rewrite the middle term and factor by grouping:
$ 2x^2 + 6x – x – 3 $
$ = 2x(x + 3) – 1(x + 3) $
$ = (2x – 1)(x + 3) $.
Example 59:
Factor $ 4x^2 + 9 $.
Solution:
This is a sum of squares, and it cannot be factored using real numbers.
Therefore, the final answer is:
$ 4x^2 + 9 $ (cannot be factored).
Example 60:
Factor $ x^3 – 64 $.
Solution:
This is a difference of cubes:
$ x^3 – 64 = (x – 4)(x^2 + 4x + 16) $.
Example 61:
Factor $ x^2 – 6x – 27 $.
Solution:
Find two numbers that multiply to -27 and add to -6. These numbers are -9 and 3.
$ x^2 – 6x – 27 = (x – 9)(x + 3) $.
Example 62:
Factor $ 8x^3 – 64x $.
Solution:
Factor out the GCF:
$ 8x(x^2 – 8) $.
Then factor the difference of squares:
$ 8x(x + 2)(x – 2) $.
Example 63:
Factor $ x^2 + 8x + 15 $.
Solution:
Find two numbers that multiply to 15 and add to 8. These numbers are 5 and 3.
$ x^2 + 8x + 15 = (x + 5)(x + 3) $.
Example 64:
Factor $ x^4 – 81 $.
Solution:
This is a difference of squares:
$ x^4 – 81 = (x^2 + 9)(x^2 – 9) $.
Then factor $ x^2 – 9 $ as a difference of squares:
$ (x^2 + 9)(x + 3)(x – 3) $.
Example 65:
Factor $ x^2 + 11x + 24 $.
Solution:
Find two numbers that multiply to 24 and add to 11. These numbers are 8 and 3.
$ x^2 + 11x + 24 = (x + 8)(x + 3) $.
Example 66:
Factor $ 25x^2 – 1 $.
Solution:
This is a difference of squares:
$ 25x^2 – 1 = (5x + 1)(5x – 1) $.
Example 67:
Factor $ 3x^2 + 10x + 8 $.
Solution:
Find two numbers that multiply to $ 3(8) = 24 $ and add to 10. These numbers are 6 and 4.
Rewrite the middle term and factor by grouping:
$ 3x^2 + 6x + 4x + 8 $
$ = 3x(x + 2) + 4(x + 2) $
$ = (3x + 4)(x + 2) $.
Example 68:
Factor $ x^2 – 12x + 36 $.
Solution:
This is a perfect square trinomial:
$ x^2 – 12x + 36 = (x – 6)^2 $.
Example 69:
Factor $ x^3 – 27 $.
Solution:
This is a difference of cubes:
$ x^3 – 27 = (x – 3)(x^2 + 3x + 9) $.
Example 70:
Factor $ 2x^2 + 5x + 3 $.
Solution:
Find two numbers that multiply to $ 2(3) = 6 $ and add to 5. These numbers are 2 and 3.
Rewrite the middle term and factor by grouping:
$ 2x^2 + 2x + 3x + 3 $
$ = 2x(x + 1) + 3(x + 1) $
$ = (2x + 3)(x + 1) $.
Example 71:
Factor $ x^2 – 9x – 22 $.
Solution:
Find two numbers that multiply to -22 and add to -9. These numbers are -11 and 2.
$ x^2 – 9x – 22 = (x – 11)(x + 2) $.
Example 72:
Factor $ 4x^2 – 9x + 2 $.
Solution:
Find two numbers that multiply to $ 4(2) = 8 $ and add to -9. These numbers are -8 and -1.
Rewrite the middle term and factor by grouping:
$ 4x^2 – 8x – x + 2 $
$ = 4x(x – 2) – 1(x – 2) $
$ = (4x – 1)(x – 2) $.
Example 73:
Factor $ 3x^2 – 27 $.
Solution:
Factor out the GCF:
$ 3(x^2 – 9) $.
Then factor $ x^2 – 9 $ as a difference of squares:
$ 3(x + 3)(x – 3) $.
Example 74:
Factor $ x^2 + 9x + 18 $.
Solution:
Find two numbers that multiply to 18 and add to 9. These numbers are 6 and 3.
$ x^2 + 9x + 18 = (x + 6)(x + 3) $.
Example 75:
Factor $ 9x^2 – 1 $.
Solution:
This is a difference of squares:
$ 9x^2 – 1 = (3x + 1)(3x – 1) $.
Example 76:
Factor $ 4x^2 + 12x + 9 $.
Solution:
Recognize this as a perfect square trinomial:
$ 4x^2 + 12x + 9 = (2x + 3)^2 $.
Example 77:
Factor $ x^2 – 4x – 21 $.
Solution:
Find two numbers that multiply to -21 and add to -4. These numbers are -7 and 3.
$ x^2 – 4x – 21 = (x – 7)(x + 3) $.
Example 78:
Factor $ x^3 + 8 $.
Solution:
This is a sum of cubes:
$ x^3 + 8 = (x + 2)(x^2 – 2x + 4) $.
Example 79:
Factor $ 5x^2 – 20x + 15 $.
Solution:
Factor out the GCF:
$ 5(x^2 – 4x + 3) $.
Then factor the trinomial:
$ 5(x – 3)(x – 1) $.
Example 80:
Factor $ x^2 + 10x + 25 $.
Solution:
Recognize this as a perfect square trinomial:
$ x^2 + 10x + 25 = (x + 5)^2 $.
Example 81:
Factor $ 9x^2 – 4 $.
Solution:
This is a difference of squares:
$ 9x^2 – 4 = (3x + 2)(3x – 2) $.
Example 82:
Factor $ 6x^2 – 19x + 10 $.
Solution:
Find two numbers that multiply to $ 6(10) = 60 $ and add to -19. These numbers are -15 and -4.
Rewrite the middle term and factor by grouping:
$ 6x^2 – 15x – 4x + 10 $
$ = 3x(2x – 5) – 2(2x – 5) $
$ = (3x – 2)(2x – 5) $.
Example 83:
Factor $ x^2 – 36 $.
Solution:
This is a difference of squares:
$ x^2 – 36 = (x + 6)(x – 6) $.
Example 84:
Factor $ 2x^3 + 54x $.
Solution:
Factor out the GCF:
$ 2x(x^2 + 27) $.
Since $ x^2 + 27 $ cannot be factored further over real numbers, the final answer is:
$ 2x(x^2 + 27) $.
Example 85:
Factor $ x^2 + 5x – 14 $.
Solution:
Find two numbers that multiply to -14 and add to 5. These numbers are 7 and -2.
$ x^2 + 5x – 14 = (x + 7)(x – 2) $.
Example 86:
Factor $ x^3 – 1 $.
Solution:
This is a difference of cubes:
$ x^3 – 1 = (x – 1)(x^2 + x + 1) $.
Example 87:
Factor $ x^2 – 5x + 6 $.
Solution:
Find two numbers that multiply to 6 and add to -5. These numbers are -3 and -2.
$ x^2 – 5x + 6 = (x – 3)(x – 2) $.
Example 88:
Factor $ x^4 – 81x^2 $.
Solution:
Factor out the common factor:
$ x^2(x^2 – 81) $.
Then factor $ x^2 – 81 $ as a difference of squares:
$ x^2(x + 9)(x – 9) $.
Example 89:
Factor $ x^2 + 8x + 16 $.
Solution:
This is a perfect square trinomial:
$ x^2 + 8x + 16 = (x + 4)^2 $.
Example 90:
Factor $ 4x^2 – 25 $.
Solution:
This is a difference of squares:
$ 4x^2 – 25 = (2x + 5)(2x – 5) $.
Example 91:
Factor $ 2x^2 – 14x + 24 $.
Solution:
Factor out the GCF:
$ 2(x^2 – 7x + 12) $.
Then factor the trinomial:
$ 2(x – 3)(x – 4) $.
Example 92:
Factor $ x^2 – 2x – 24 $.
Solution:
Find two numbers that multiply to -24 and add to -2. These numbers are -6 and 4.
$ x^2 – 2x – 24 = (x – 6)(x + 4) $.
Example 93:
Factor $ 3x^2 + 12x + 12 $.
Solution:
Factor out the GCF:
$ 3(x^2 + 4x + 4) $.
Recognize this as a perfect square trinomial:
$ 3(x + 2)^2 $.
Example 94:
Factor $ 16x^2 – 64 $.
Solution:
Factor out the GCF:
$ 16(x^2 – 4) $.
Then factor $ x^2 – 4 $ as a difference of squares:
$ 16(x + 2)(x – 2) $.
Example 95:
Factor $ x^2 – 18x + 45 $.
Solution:
Find two numbers that multiply to 45 and add to -18. These numbers are -15 and -3.
$ x^2 – 18x + 45 = (x – 15)(x – 3) $.
Example 96:
Factor $ 4x^2 – 9 $.
Solution:
This is a difference of squares:
$ 4x^2 – 9 = (2x + 3)(2x – 3) $.
Example 97:
Factor $ x^3 + 27x $.
Solution:
Factor out the GCF:
$ x(x^2 + 27) $.
Since $ x^2 + 27 $ cannot be factored further over real numbers, the final answer is:
$ x(x^2 + 27) $.
Example 98:
Factor $ x^2 + 3x – 4 $.
Solution:
Find two numbers that multiply to -4 and add to 3. These numbers are 4 and -1.
$ x^2 + 3x – 4 = (x + 4)(x – 1) $.
Example 99:
Factor $ 4x^2 + 25 $.
Solution:
This is a sum of squares, and it cannot be factored using real numbers.
Therefore, the final answer is:
$ 4x^2 + 25 $ (cannot be factored).
Example 100:
Factor $ x^3 – 125 $.
Solution:
This is a difference of cubes:
$ x^3 – 125 = (x – 5)(x^2 + 5x + 25) $.
