SOLUTION: If the product of the binomial (2x-5) with the trinomial (3x^2+2x-5) is formed, what is the coefficient of the x^2 term? Please help me answer this.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: If the product of the binomial (2x-5) with the trinomial (3x^2+2x-5) is formed, what is the coefficient of the x^2 term? Please help me answer this.       Log On


   

Question 1123580: If the product of the binomial (2x-5) with the trinomial (3x^2+2x-5) is formed, what is the coefficient of the x^2 term? Please help me answer this.
Found 2 solutions by math_helper, greenestamps:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at +%283x%5E2%2B2x-5%29%282x-5%29+ there will be two multiplications that produce an +x%5E2 term:
+%283x%5E2%29%2A%28-5%29+ + +%282x%29%2A%282x%29+
= +-15x%5E2+ + +4x%5E2+
= +-11x%5E2+
so the coefficient is +highlight+%28+-11+%29+

You know this because the exponents of x must add to 2 (2+0=2, 1+1=2, and there are no others).
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The long way would be to multiply out then combine like terms, and then pick the x%5E2+ coefficient:
+%283x%5E2%2B2x-5%29%282x-5%29+
= +%283x%5E2%2B2x-5%29%282x%29+%2B+%283x%5E2%2B2x-5%29%28-5%29+
= +%286x%5E3+%2B+4x%5E2+-+10x%29+%2B+%28-15x%5E2+-+10x+%2B+25%29+
= +6x%5E3+-+11x%5E2+-+20x+%2B+25+
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EDITED to make long way answer to be 100% correct (had +5 in trinomial, corrected it to be -5)

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


It wouldn't take much work to perform the entire polynomial multiplication; however, there are times when it is a useful skill to be able to determine the coefficient of a particular power in the product of two (or more) binomials, without performing the entire multiplication.

To do this, you need to think about where terms of particular powers in the product come from in the multiplication. In this product....

The constant term is the product of the constants in the two polynomial factors: (-5)*(-5) = 25

The leading (x^3) coefficient is the product of the leading coefficients of the two factors: (2)*(3) = 6

The coefficient of the x term in the product comes from two places -- the product of the x term in the first polynomial and the constant in the second; and the product of the constant in the first polynomial and the x term in the other: (2)(-5)+(-5)(2) = -20

See if you can find the answer to your problem using this process.

There are two places where the product of one term of each polynomial will produce an x^2 term.
What are those two places?
What are the coefficients of those two products?
The answer to your question is the sum of those two coefficients.