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Chapter 6: Problem 113

Factor completely. $$10 x^{2}(x+1)-7 x(x+1)-6(x+1)$$

Short Answer

Expert verified
The complete factorization of the expression \(10x^2(x+1) - 7x(x+1) - 6(x+1)\) is \((x+1)(2x + 1)(5x - 6)\).

Step by step solution

01

Identify the common factor

Observe the expression and identify the common factor. Here, we can see that each term is multiplied by \((x+1)\) so that is our common factor.
02

Factor out the common factor

We can factor out \((x+1)\) from each term: \(10x^2(x+1) - 7x(x+1) - 6(x+1) = (10x^2 - 7x - 6)(x+1)\).
03

Factor the quadratic expression

Now the problem is reduced to factoring the quadratic expression \(10x^2 - 7x - 6\). This is a trinomial with a non-one leading coefficient. We continue factoring by looking for two numbers that multiply to \(10*(-6) = -60\) and add to \(-7\). The numbers \(-12\) and \(5\) fulfil these criteria: \(10x^2 - 7x - 6 = 10x^2 - 12x + 5x - 6.\)
04

Regroup and factor by grouping

Next, regroup and use factoring-by-grouping: \(10x^2 - 12x + 5x - 6 = 2x(5x - 6) + 1(5x - 6). Now, we have a common factor of \(5x - 6\) to factor out: 2x(5x - 6) + 1(5x - 6) = (2x + 1)(5x - 6).
05

Combine the factors

Finally, we combine our factors to express the original expression as a product of its simplest factors: \(10x^2(x+1) - 7x(x+1) - 6(x+1) = (x+1)(2x + 1)(5x - 6).

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Most popular questions from this chapter

Determine whether each statement 鈥渕akes sense鈥� or 鈥渄oes not make sense鈥� and explain your reasoning. It takes a great deal of practice to get good at factoring a wide variety of polynomials.

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$y^{3}+2 y^{2}-y-2$$

Contain polynomials in several variables. Factor each polynomial completely and check using multiplication. $$16 a^{3} b^{2}-4 a b^{2}$$

Factor completely. $$4 x^{4}-9 x^{2}+5$$

Now let's move on to factorizations that may require two or more techniques. Factor completely, or state that the polynomial is prime. Check factorizations using multiplication or a graphing utility. $$9 y^{2}+64$$
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