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Question

Difficulty: MediumFactoring Polynomials

Simplify the given algebraic expression by factoring it completely, then determine the missing positive integer values.

Answer:The completely factored form of the expression 2y2−10y−282y^2 - 10y - 28 can be written in the format 2(y−a)(y+b)2(y - a)(y + b), where aa and bb are positive integers.

The value of aa is 【7】 and the value of bb is 【2】.

Answer

The value of a is 7 and the value of b is 2.
Factoring out the GCF of 2 yields 2(y2−5y−14)2(y^2 - 5y - 14). The trinomial y2−5y−14y^2 - 5y - 14 factors into (y−7)(y+2)(y - 7)(y + 2). Matching this with the given structure 2(y−a)(y+b)2(y - a)(y + b) shows that the positive integers are a=7a = 7 and b=2b = 2.

Step-by-Step Solution

1
Factor out the greatest common factor (GCF) from all terms in the polynomial.
2(y2−5y−14)2(y^2 - 5y - 14)
Identifying and factoring out the GCF of 2 simplifies the polynomial, revealing a trinomial with a leading coefficient of 1, which is easier to factor.
2
Factor the resulting quadratic trinomial y2−5y−14y^2 - 5y - 14.
The binomial factors are (y−7)(y - 7) and (y+2)(y + 2).
To factor y2−5y−14y^2 - 5y - 14, we need to find two numbers that multiply to the constant term (-14) and add up to the middle term's coefficient (-5). The numbers -7 and 2 satisfy both conditions (-7 * 2 = -14 and -7 + 2 = -5).
3
Write the complete factorization and match it to the provided template 2(y−a)(y+b)2(y - a)(y + b).
The expression is 2(y−7)(y+2)2(y - 7)(y + 2). Matching this to the template gives a=7a = 7 and b=2b = 2.
Comparing the forms directly, the term subtracted from y is 7 (so a = 7) and the term added to y is 2 (so b = 2).

Key Concept

Factoring polynomials using the Greatest Common Factor (GCF) and factoring quadratic trinomials.
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Topics

Factoring Polynomials

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