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Difficulty: EasyFactoring Polynomials

The trinomial 2x2+7x+32x^2 + 7x + 3 can be factored into the product of two binomials of the form (2x+a)(x+b)(2x + a)(x + b), where aa and bb are integers. What is the value of the expression a+2ba + 2b?

Answer: 7

Answer

The value of the expression a+2ba + 2b is 7.
Expanding the factored template (2x+a)(x+b)(2x + a)(x + b) yields 2x2+(a+2b)x+ab2x^2 + (a + 2b)x + ab. Comparing this to the given expression 2x2+7x+32x^2 + 7x + 3, the coefficient of xx on the left side is a+2ba + 2b, and on the right side is 7. Therefore, a+2b=7a + 2b = 7. Alternatively, factoring 2x2+7x+32x^2 + 7x + 3 yields (2x+1)(x+3)(2x + 1)(x + 3), where a=1a = 1 and b=3b = 3. Substituting these integers into a+2ba + 2b gives 1+2(3)=71 + 2(3) = 7.

Step-by-Step Solution

1
Expand the expression (2x+a)(x+b)(2x + a)(x + b) using the FOIL method.
2x2+2bx+ax+ab=2x2+(a+2b)x+ab2x^2 + 2bx + ax + ab = 2x^2 + (a + 2b)x + ab
Expanding the template allows direct comparison of its coefficients with the given trinomial.
2
Equate the coefficients of the expanded template to the given trinomial 2x2+7x+32x^2 + 7x + 3.
a+2b=7a + 2b = 7 and ab=3ab = 3
For the two polynomial expressions to be equivalent for all values of xx, their corresponding coefficients must be equal.
3
Identify the requested value directly from the system of equations.
7
The question asks for the value of a+2ba + 2b, which is precisely the coefficient of the linear xx term.

Key Concept

Factoring quadratic trinomials with a leading coefficient greater than 1

Alternative Method

Factor the trinomial 2x2+7x+32x^2 + 7x + 3 using the AC method: multiply the leading coefficient (2) and the constant term (3) to get 6. Find two numbers that multiply to 6 and add to 7, which are 6 and 1. Rewrite the middle term: 2x2+6x+x+32x^2 + 6x + x + 3. Factor by grouping: 2x(x+3)+1(x+3)=(2x+1)(x+3)2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3). Compare this to (2x+a)(x+b)(2x + a)(x + b) to find a=1a = 1 and b=3b = 3, then compute a+2b=1+2(3)=7a + 2b = 1 + 2(3) = 7.
Estimated Time:45s
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Topics

Factoring Polynomials

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