Factor and simplify each algebraic expression.

Question

Factor and simplify each algebraic expression. $-8(4x + 3)^{-2} + 10(5x + 1)(4x + 3)^{-1}$

Accepted Answer

Hello, today we're gonna be simplifying the following algebraic expression. So what we are given is negative six, multiplied by the quantity two X plus one raced to the power of negative four plus 15, multiplied by five X plus two, multiplied by two X plus one race to the power of negative three. Now, the first step into simplifying this expression is to change the negative exponents into positive exponents. In order to do this, we can take the reciprocal of the quantities with the negative exponents. And that will allow us to rewrite the negative exponents as positive exponents. So if we take the reciprocals, we can rewrite the expression as negative six over the quantity two X plus one race to the power of four plus 15, multiplied by five X plus two, all over two X plus one raced to the power of positive three. Now, what we wanna do is you want to combine the expression into a single statement. In order to do that, we need to get the same denominator in both of the fractions. We have two X plus one race to the power of four and we have two X plus one raised to the power of three, we can increase this power of three by one by multiplying two X plus one raised to the power three by another quantity of two X plus one. But if we multiply the denominator on the right hand side by two X plus one, we must also multiply the numerator by two X plus one. We can rewrite the right hand side of the equation as negative 6/2 X plus one race to the power of four plus 15, multiplied by five X plus two, multiplied by two X plus one, all over two X plus one race to the power of four. Now that we have the same denominator in both of these fractions, we can combine everything into a single fraction. And that will allow us to rewrite the expression as negative six plus 15, multiplied by the quantity five X plus two, multiplied by two X plus one all over two X plus one raised to the power of positive form. Now, what we need to do is we need to simplify the numerator in order to do this, we're going to need to foil the two quantities in the numerator together. So let's go ahead and foil five X plus two and two X plus one together. In order to do this, we're going to distribute the first term in the first group to each term in the second group. So five X multiplied by two X is going to give us 10 X squared and five X multiplied by one is going to give us five X. Then we're gonna repeat this process with the second term in the first group and distribute it to each term in the second group. So two multiplied by two X is going to give us four X and two multiplied by one is going to give us two. Finally, we're going to combine the light terms together, which is five X and four X. And this is going to simplify to give us 10 X squared plus nine X plus two. So this is going to be the foiled form of the two quantities. And if we put this back into the numerator, we get negative six plus 15 multiplied by the quantity 10 X squared plus nine X plus two all over two X plus one race to the power of four. Now, what we need to do in order to simplify the numerator is to distribute the value of into the trinomial. So 15 multiplied by 10 X squared is going to give us 150 X squared. 15 multiplied by nine X is going to give us X and 15 multiplied by two is going to give us 30. So we can rewrite the numerator as negative six plus 100 and X squared plus 1, 35 X plus 30 all over two X plus one raised to the power of four. Next, we can go ahead and combine the constants which are going to be negative six and positive 30. And that is going to give us 150 X squared plus 135 X plus 24 all over two X plus one raised to the power of four. Now, there's one more thing to notice about the numerator, we have the coefficients of 151 24. All of these coefficients are divisible by three. So we can go ahead and factor out a three from the numerator. And if we factor the three, we can rewrite the expression as three times the quantity X squared plus 45 X plus eight all over the quantity of two X plus one raised to the power of four. And this is going to be the simplified form of the algebraic expression. And with that being said, the answer to this problem is going to be B so I hope this video helps you in understanding how to approach this problem. And I'll go ahead and see you all in the next video.

Factor and simplify each algebraic expression. $-8(4x + 3)^{-2} + 10(5x + 1)(4x + 3)^{-1}$

Key Concepts

Negative Exponents

Factoring Algebraic Expressions

Combining Like Terms with Rational Expressions

Watch next

Factor and simplify each algebraic expression. −8(4x+3)−2+10(5x+1)(4x+3)−1-8(4x + 3)^{-2} + 10(5x + 1)(4x + 3)^{-1}−8(4x+3)−2+10(5x+1)(4x+3)−1

Key Concepts

Negative Exponents

Factoring Algebraic Expressions

Combining Like Terms with Rational Expressions

Watch next

Factor and simplify each algebraic expression. $-8(4x + 3)^{-2} + 10(5x + 1)(4x + 3)^{-1}$