Factoring Polynomials Notes and Worksheets - Lindsay Bowden - Free Printable

To solve the problem of factoring each trinomial completely, we will use the method of factoring by grouping or by finding two numbers that multiply to the product of the leading coefficient and the constant term and add up to the middle coefficient. Let's go through each problem step by step.

Problem 1: \(4x^2 + 4x - 35\)

We need to find two numbers that multiply to \(4 \cdot (-35) = -140\) and add up to \(4\). These numbers are \(14\) and \(-10\).
\[
4x^2 + 4x - 35 = 4x^2 + 14x - 10x - 35
\]
Now, factor by grouping:
\[
= 2x(2x + 7) - 5(2x + 7) = (2x - 5)(2x + 7)
\]
So, the factored form is:
\[
\boxed{(2x - 5)(2x + 7)}
\]

Problem 2: \(5x^2 + 37x - 24\)

We need to find two numbers that multiply to \(5 \cdot (-24) = -120\) and add up to \(37\). These numbers are \(40\) and \(-3\).
\[
5x^2 + 37x - 24 = 5x^2 + 40x - 3x - 24
\]
Now, factor by grouping:
\[
= 5x(x + 8) - 3(x + 8) = (5x - 3)(x + 8)
\]
So, the factored form is:
\[
\boxed{(5x - 3)(x + 8)}
\]

Problem 3: \(3x^2 + 12x + 12\)

First, factor out the greatest common factor (GCF), which is \(3\):
\[
3x^2 + 12x + 12 = 3(x^2 + 4x + 4)
\]
Now, factor the quadratic expression inside the parentheses:
\[
x^2 + 4x + 4 = (x + 2)^2
\]
So, the factored form is:
\[
\boxed{3(x + 2)^2}
\]

Problem 4: \(6x^2 - 44x + 14\)

First, factor out the GCF, which is \(2\):
\[
6x^2 - 44x + 14 = 2(3x^2 - 22x + 7)
\]
Now, factor the quadratic expression inside the parentheses. We need to find two numbers that multiply to \(3 \cdot 7 = 21\) and add up to \(-22\). These numbers are \(-21\) and \(-1\).
\[
3x^2 - 22x + 7 = 3x^2 - 21x - x + 7
\]
Now, factor by grouping:
\[
= 3x(x - 7) - 1(x - 7) = (3x - 1)(x - 7)
\]
So, the factored form is:
\[
\boxed{2(3x - 1)(x - 7)}
\]

Problem 5: \(3x^2 - 23x - 36\)

We need to find two numbers that multiply to \(3 \cdot (-36) = -108\) and add up to \(-23\). These numbers are \(-27\) and \(4\).
\[
3x^2 - 23x - 36 = 3x^2 - 27x + 4x - 36
\]
Now, factor by grouping:
\[
= 3x(x - 9) + 4(x - 9) = (3x + 4)(x - 9)
\]
So, the factored form is:
\[
\boxed{(3x + 4)(x - 9)}
\]

Problem 6: \(2x^2 - 3x + 1\)

We need to find two numbers that multiply to \(2 \cdot 1 = 2\) and add up to \(-3\). These numbers are \(-2\) and \(-1\).
\[
2x^2 - 3x + 1 = 2x^2 - 2x - x + 1
\]
Now, factor by grouping:
\[
= 2x(x - 1) - 1(x - 1) = (2x - 1)(x - 1)
\]
So, the factored form is:
\[
\boxed{(2x - 1)(x - 1)}
\]

Problem 7: \(20x^2 + 2x - 6\)

First, factor out the GCF, which is \(2\):
\[
20x^2 + 2x - 6 = 2(10x^2 + x - 3)
\]
Now, factor the quadratic expression inside the parentheses. We need to find two numbers that multiply to \(10 \cdot (-3) = -30\) and add up to \(1\). These numbers are \(6\) and \(-5\).
\[
10x^2 + x - 3 = 10x^2 + 6x - 5x - 3
\]
Now, factor by grouping:
\[
= 2x(5x + 3) - 1(5x + 3) = (2x - 1)(5x + 3)
\]
So, the factored form is:
\[
\boxed{2(2x - 1)(5x + 3)}
\]

Problem 8: \(3x^2 + 25x + 42\)

We need to find two numbers that multiply to \(3 \cdot 42 = 126\) and add up to \(25\). These numbers are \(14\) and \(11\).
\[
3x^2 + 25x + 42 = 3x^2 + 14x + 11x + 42
\]
Now, factor by grouping:
\[
= x(3x + 14) + 3(3x + 14) = (x + 3)(3x + 14)
\]
So, the factored form is:
\[
\boxed{(x + 3)(3x + 14)}
\]

Problem 9: \(2x^2 - 3x - 20\)

We need to find two numbers that multiply to \(2 \cdot (-20) = -40\) and add up to \(-3\). These numbers are \(-8\) and \(5\).
\[
2x^2 - 3x - 20 = 2x^2 - 8x + 5x - 20
\]
Now, factor by grouping:
\[
= 2x(x - 4) + 5(x - 4) = (2x + 5)(x - 4)
\]
So, the factored form is:
\[
\boxed{(2x + 5)(x - 4)}
\]

Problem 10: \(35x^2 - 11x - 6\)

We need to find two numbers that multiply to \(35 \cdot (-6) = -210\) and add up to \(-11\). These numbers are \(-21\) and \(10\).
\[
35x^2 - 11x - 6 = 35x^2 - 21x + 10x - 6
\]
Now, factor by grouping:
\[
= 7x(5x - 3) + 2(5x - 3) = (7x + 2)(5x - 3)
\]
So, the factored form is:
\[
\boxed{(7x + 2)(5x - 3)}
\]

The final answers are:
\[
\boxed{(2x - 5)(2x + 7), (5x - 3)(x + 8), 3(x + 2)^2, 2(3x - 1)(x - 7), (3x + 4)(x - 9), (2x - 1)(x - 1), 2(2x - 1)(5x + 3), (x + 3)(3x + 14), (2x + 5)(x - 4), (7x + 2)(5x - 3)}
\]