QuestionFactor completely. $16 x^{2}-40 x-56$

Studdy Solution

STEP 1

1. The expression $16x^2 - 40x - 56$ is a quadratic polynomial.
2. The expression can be factored completely over the integers.

STEP 2

1. Factor out the greatest common factor (GCF) from the expression.
2. Factor the resulting quadratic expression completely.

STEP 3

Identify the greatest common factor (GCF) of the terms in the expression $16x^2 - 40x - 56$ .
The GCF of 16, 40, and 56 is 8.
Factor out the GCF:
$16x^2 - 40x - 56 = 8(2x^2 - 5x - 7)$

STEP 4

Now, focus on factoring the quadratic expression $2x^2 - 5x - 7$ .
To factor this, we need two numbers that multiply to $2 \times (-7) = -14$ and add to $-5$ .
The numbers that satisfy these conditions are $-7$ and $2$ .
Rewrite the middle term $-5x$ using $-7$ and $2$ :
$2x^2 - 5x - 7 = 2x^2 - 7x + 2x - 7$

STEP 5

Factor by grouping:
Group the terms: $(2x^2 - 7x) + (2x - 7)$ .
Factor out the common factors in each group:
$2x(x - \frac{7}{2}) + 1(x - 7)$
Notice that the expression should be:
$2x(x - 3.5) + 1(x - 7)$
However, let's correct and simplify:
$(2x + 1)(x - 7)$

STEP 6

Combine the factored terms with the GCF:
$8(2x + 1)(x - 7)$
This is the completely factored form of the original expression.

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