Answer :
To determine which expression is equal to [tex]\((3x - 5)(2x - 7)\)[/tex], let's break down the multiplication step by step.
First, use the distributive property (also known as the FOIL method for binomials) to expand the expression:
1. First terms: Multiply the first terms of each binomial.
[tex]\[
3x \cdot 2x = 6x^2
\][/tex]
2. Outer terms: Multiply the outer terms of the binomials.
[tex]\[
3x \cdot (-7) = -21x
\][/tex]
3. Inner terms: Multiply the inner terms of the binomials.
[tex]\[
-5 \cdot 2x = -10x
\][/tex]
4. Last terms: Multiply the last terms of the binomials.
[tex]\[
-5 \cdot (-7) = 35
\][/tex]
Next, combine all these products:
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]
Now, combine like terms (the [tex]\(x\)[/tex] terms):
[tex]\[
-21x - 10x = -31x
\][/tex]
So, the expanded form is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
Therefore, the expression [tex]\((3x - 5)(2x - 7)\)[/tex] is equal to [tex]\(6x^2 - 31x + 35\)[/tex].
Among the given options, the correct expression is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
First, use the distributive property (also known as the FOIL method for binomials) to expand the expression:
1. First terms: Multiply the first terms of each binomial.
[tex]\[
3x \cdot 2x = 6x^2
\][/tex]
2. Outer terms: Multiply the outer terms of the binomials.
[tex]\[
3x \cdot (-7) = -21x
\][/tex]
3. Inner terms: Multiply the inner terms of the binomials.
[tex]\[
-5 \cdot 2x = -10x
\][/tex]
4. Last terms: Multiply the last terms of the binomials.
[tex]\[
-5 \cdot (-7) = 35
\][/tex]
Next, combine all these products:
[tex]\[
6x^2 - 21x - 10x + 35
\][/tex]
Now, combine like terms (the [tex]\(x\)[/tex] terms):
[tex]\[
-21x - 10x = -31x
\][/tex]
So, the expanded form is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
Therefore, the expression [tex]\((3x - 5)(2x - 7)\)[/tex] is equal to [tex]\(6x^2 - 31x + 35\)[/tex].
Among the given options, the correct expression is:
[tex]\[
6x^2 - 31x + 35
\][/tex]
