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Math

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PROBLEM

Factor the trinomial 42x253x+1542 x^{2}-53 x+15.

STEP 1

Assumptions1. The given trinomial is 42x53x+1542x^{} -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 pp and qq. So we havep+q=53p + q = -53pq=4215p \cdot q =42 \cdot15

STEP 3

Calculate the product of the coefficient of the first term and the last term.
pq=4215=630p \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.
42x253x+15=42x221x30x+1542x^{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.
42x221x30x+15=21x(2x1)15(2x1)42x^{2} -21x -30x +15 =21x(2x -1) -15(2x -1)

STEP 7

Notice that (2x1)(2x -1) is a common factor. Factor out (2x1)(2x -1).
21x(2x1)15(2x1)=(2x1)(21x15)21x(2x -1) -15(2x -1) = (2x -1)(21x -15)

SOLUTION

The factored form of the trinomial 42x253x+1542x^{2} -53x +15 is (2x1)(21x15)(2x -1)(21x -15).
The trinomial 42x253x+1542x^{2} -53x +15 factors to (2x1)(21x15)(2x -1)(21x -15).

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