Factor $x^3-4 x^2+7 x-28$ by grouping. What is the resulting expression?
A. $\left(x^2-4\right)(x+7)$
B. $\left(x^2+4\right)(x-7)$
C. $\left(x^2-7\right)(x+4)$
D. $\left(x^2+7\right)(x-4)$
Answer (2)
The polynomial x 3 − 4 x 2 + 7 x − 28 can be factored by grouping. After factoring, we find that it equals ( x 2 + 7 ) ( x − 4 ) , which corresponds to option D. Therefore, the correct answer is D: ( x 2 + 7 ) ( x − 4 ) .
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Group the terms: ( x 3 − 4 x 2 ) + ( 7 x − 28 ) .
Factor out the greatest common factor: x 2 ( x − 4 ) + 7 ( x − 4 ) .
Factor out the common binomial: ( x 2 + 7 ) ( x − 4 ) .
The factored expression is ( x 2 + 7 ) ( x − 4 ) .
Explanation
Understanding the problem We are given the expression x 3 − 4 x 2 + 7 x − 28 and we want to factor it by grouping.
Grouping terms We group the first two terms and the last two terms: ( x 3 − 4 x 2 ) + ( 7 x − 28 ) .
Factoring out common factors We factor out the greatest common factor from each group: x 2 ( x − 4 ) + 7 ( x − 4 ) .
Factoring out the common binomial We factor out the common binomial factor ( x − 4 ) : ( x 2 + 7 ) ( x − 4 ) .
Final Answer The factored expression is ( x 2 + 7 ) ( x − 4 ) . Comparing this with the given options, we find that the correct factorization is ( x 2 + 7 ) ( x − 4 ) .
Examples
Factoring by grouping is a useful technique in algebra. For example, suppose you want to find the dimensions of a rectangular prism whose volume is given by the expression x 3 − 4 x 2 + 7 x − 28 . By factoring this expression into ( x 2 + 7 ) ( x − 4 ) , you can determine possible expressions for the dimensions of the prism. One possible set of dimensions could be x 2 + 7 and x − 4 , with a height of 1. This technique is also used in more advanced mathematical problems, such as finding roots of polynomials.
