QuestionFactor the trinomial $42 x^{2}-53 x+15$ .

Studdy Solution

STEP 1

Assumptions1. The given trinomial is $42x^{} -53x +15$ . . We need to factorize the trinomial into two binomials.

STEP 2

To factorize the trinomial, we need to find two numbers that add up to the coefficient of the middle term (-53) and multiply to give the product of the coefficient of the first term (42) and the last term (15).
Let's denote these two numbers as $p$ and $q$ . So we have $p + q = -53$ $p \cdot q =42 \cdot15$

STEP 3

Calculate the product of the coefficient of the first term and the last term.
$p \cdot q =42 \cdot15 =630$

STEP 4

Now, we need to find two numbers that add up to -53 and multiply to630. These numbers are -21 and -30.

STEP 5

Rewrite the middle term of the trinomial as the sum of -21x and -30x.
$42x^{2} -53x +15 =42x^{2} -21x -30x +15$

STEP 6

Factor by grouping. Group the first two terms together and the last two terms together.
$42x^{2} -21x -30x +15 =21x(2x -1) -15(2x -1)$

STEP 7

Notice that $(2x -1)$ is a common factor. Factor out $(2x -1)$ .
$21x(2x -1) -15(2x -1) = (2x -1)(21x -15)$

STEP 8

The factored form of the trinomial $42x^{2} -53x +15$ is $(2x -1)(21x -15)$ .
The trinomial $42x^{2} -53x +15$ factors to $(2x -1)(21x -15)$ .

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