x²(x + 4) + 5(x + 4)
Factoring (also called factorization) is the reserve operation of multiplication. When we factor (or factorize) a polynomial, we write it as a product of its factors.
Grouping is one of the factoring methods. Other methods such as isolating common factors and identities.
Steps for grouping:
Let’s start.
[tex]\boxed{ \ x^3 + 4x^2 + 5x + 20 = \ ? \ }[/tex]
[tex]\boxed{ \ = (x^3 + 4x^2) + (5x + 20) \ }[/tex]
[tex]\boxed{\boxed{ \ = x^2(x + 4) + 5(x + 4) \ }}[/tex]
We can also group the terms as follows.
[tex]\boxed{ \ x^3 + 4x^2 + 5x + 20 = \ ? \ }[/tex]
[tex]\boxed{ \ = (x^3 + 5x) + (4x^2 + 20) \ }[/tex]
[tex]\boxed{\boxed{ \ = x(x^2 + 5) + 4(x^2 + 5) \ }}[/tex]
We will get the same results from the two ways above. Let's see.
Thus the part which shows one way to determine the factors of x³ + 4x² + 5x + 20 by grouping is [tex]\boxed{\boxed{ \ x^2(x + 4) + 5(x + 4) \ }}[/tex]
Keywords: which shows, one way, to determine, the factors of x³ + 4x² + 5x + 20, by grouping, common factor, polynomial, identities
The correct option is [tex]\fbox{\text{Option B}}[/tex] that is [tex]{x^2}\left( {x + 4} \right) + 5\left( {x + 4} \right)[/tex] that shows one way to determine the factors of [tex]{x^3} + 4{x^2} + 5x + 20[/tex].
Further explanation:
Factors are the numbers if we multiply them we get the original number.
Factorization is finding the numbers if we multiply them we get the original number.
Prime factorization is finding the numbers if we multiply them we get the original number.
Given:
The polynomial is [tex]{x^3} + 4{x^2} + 5x + 20[/tex].
The options are,
A. [tex]x\left( {{x^2} + 4} \right) + 5\left( {{x^2} + 4} \right)[/tex].
B. [tex]{x^2}\left( {x + 4} \right) + 5\left( {x + 4} \right)[/tex].
C. [tex]{x^2}\left( {x + 5} \right) + 5\left( {x + 5} \right)[/tex].
D. [tex]x\left( {{x^2} + 5} \right) + 4x\left( {{x^2} + 5} \right)[/tex]
Explanation:
Consider the polynomial [tex]{x^3} + 4{x^2} + 5x + 20[/tex] as [tex]P\left( x \right)[/tex].
Steps involve in finding the factors of [tex]{x^3} + 4{x^2} + 5x + 20[/tex] are as follows,
First we have to make the groups of the terms as,
[tex]P\left( x \right) = \left( {{x^3} + 4{x^2}} \right) + \left( {5x + 20} \right)[/tex]
Now factorize the above 2 groups.
[tex]\begin{gathered}P\left( x \right) = {x^2}\left( {x + 4} \right)+5\left( {{x^2}+4} \right)\\= \left( {{x^2}+5} \right)\left( {x+4} \right) \\\end{gathered}[/tex]
The factors of [tex]{x^3} + 4{x^2} + 5x + 20\text{ } \text{are} \left( {{x^2} + 5} \right) \text{and} \left( {x + 4} \right)[/tex].
Option A is not correct as the first common factor is [tex]x[/tex] from group one but it not the highest common factor.
[tex]\fbox{\text{Option B}}[/tex] is correct as the factors are same as the factors of the polynomial.
Option C is not correct as the factors are not same as the factors of the polynomial.
Option D is not correct as the factors are not same as the factors of the polynomial.
Learn more:
1. Learn more about the polynomial https://brainly.com/question/12996944
2. Learn more about logarithm model https://brainly.com/question/13005829
3. Learn more about the product of binomial and trinomial https://brainly.com/question/1394854
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Polynomials
Keywords: factor, factorization, polynomial, quadratic, cubic, greatest common factor, groups, multiplication, product, identities, common factor, expression, terms, grouping.
