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Answer :
Sure! Let's go through the steps to factor the polynomial [tex]\( 4x^3 - 20x^2 + 9x - 45 \)[/tex] by grouping.
1. Group the terms in pairs:
[tex]$$ (4x^3 - 20x^2) + (9x - 45) $$[/tex]
2. Factor out the greatest common factor (GCF) from each pair:
- For the first group [tex]\( 4x^3 - 20x^2 \)[/tex], the GCF is [tex]\( 4x^2 \)[/tex]:
[tex]$$ 4x^2(x - 5) $$[/tex]
- For the second group [tex]\( 9x - 45 \)[/tex], the GCF is [tex]\( 9 \)[/tex]:
[tex]$$ 9(x - 5) $$[/tex]
3. Rewrite the expression using these factors:
[tex]$$ 4x^2(x - 5) + 9(x - 5) $$[/tex]
4. Factor out the common binomial factor [tex]\( (x - 5) \)[/tex]:
[tex]$$ (x - 5)(4x^2 + 9) $$[/tex]
So, the factored form of the polynomial [tex]\( 4x^3 - 20x^2 + 9x - 45 \)[/tex] is:
[tex]$$ (x - 5)(4x^2 + 9) $$[/tex]
This is the complete factorization of the given polynomial by grouping.
1. Group the terms in pairs:
[tex]$$ (4x^3 - 20x^2) + (9x - 45) $$[/tex]
2. Factor out the greatest common factor (GCF) from each pair:
- For the first group [tex]\( 4x^3 - 20x^2 \)[/tex], the GCF is [tex]\( 4x^2 \)[/tex]:
[tex]$$ 4x^2(x - 5) $$[/tex]
- For the second group [tex]\( 9x - 45 \)[/tex], the GCF is [tex]\( 9 \)[/tex]:
[tex]$$ 9(x - 5) $$[/tex]
3. Rewrite the expression using these factors:
[tex]$$ 4x^2(x - 5) + 9(x - 5) $$[/tex]
4. Factor out the common binomial factor [tex]\( (x - 5) \)[/tex]:
[tex]$$ (x - 5)(4x^2 + 9) $$[/tex]
So, the factored form of the polynomial [tex]\( 4x^3 - 20x^2 + 9x - 45 \)[/tex] is:
[tex]$$ (x - 5)(4x^2 + 9) $$[/tex]
This is the complete factorization of the given polynomial by grouping.
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