Rewriting Polynomials by Factoring

Let's start by factoring out the greatest common factor. Then, we will factor the resulting trinomial.

Factor Out the Greatest Common Factor

The greatest common factor is a common factor of all the terms in the expression. It is the common factor with the greatest coefficient and the greatest exponent. In this case, the greatest common factor is

x^{2} .

2 x^{4} + 7 x^{3} + 5 x^{2}

SplitIntoFactorsSplit into factors

x^{2} \cdot 2 x^{2} + x^{2} \cdot 7 x + x^{2} \cdot 5

FactorOutFactor out

x^{2}

x^{2} (2 x^{2} + 7 x + 5)

The result of factoring out the greatest common factor from the given expression is a quadratic expression with a leading coefficient of

2 .

x^{2} (2 x^{2} + 7 x + 5)

Let's temporarily only focus on the expression in parentheses, and bring back the greatest common factor after factoring.

Factor the Quadratic Trinomial

Here we have a quadratic trinomial of the form $a x^{2} + b x + c,$ where $∣ a ∣ \neq = 1$ and there are no common factors. To factor this expression, we will rewrite the middle term, $b x,$ as two terms. The coefficients of these two terms will be factors of $a c$ whose sum must be $b .$ $2 x^{2} + 7 x + 5$ We have that $a = 2,$ $b = 7,$ and $c = 5 .$ There are now three steps we need to follow in order to rewrite the above expression.

Find $a c .$ Since we have that $a = 2$ and $c = 5,$ the value of $a c$ is $2 \times 5 = 10 .$
Find factors of $a c$ . Since $a c = 10,$ which is positive, we need factors of $a c$ to have the same sign — both positive or both negative — in order for the product to be positive. Since $b = 7,$ which is also positive, those factors will need to be positive so that their sum is positive.

$\underline{1^{st} factor} - 2 2 \underline{2^{nd} factor} - 5 5 \underline{Sum} - 2 + (- 5) 2 + 5 \underline{Result} - 7 7$

Rewrite $b x$ as two terms. Now that we know which factors are the ones to be used, we can rewrite $b x$ as two terms. $2 x^{2} + 7 x + 5 \Leftrightarrow 2 x^{2} + 2 x + 5 x + 5$

Finally, we will factor the last expression obtained.

2 x^{2} + 2 x + 5 x + 5

FactorOutFactor out

2 x

2 x (x + 1) + 5 x + 5

FactorOutFactor out

5

2 x (x + 1) + 5 (x + 1)

FactorOutFactor out

(x + 1)

(x + 1) (2 x + 5)

We have factored the quadratic trinomial.

2 x^{2} + 7 x + 5 \Leftrightarrow (x + 1) (2 x + 5)

Finally, we need to include the greatest common factor that we factored out at the beginning.

2 x^{4} + 7 x^{3} + 5 x^{2} \Leftrightarrow x^{2} (x + 1) (2 x + 5)

Rewriting Polynomials by Factoring 1.5 - Solution

Factor Out the Greatest Common Factor

Factor the Quadratic Trinomial