Answer :
Sure, let's factor the trinomial [tex]\(15x^2 + 46xy + 35y^2\)[/tex] completely by grouping.
1. Identify the trinomial:
We have [tex]\(15x^2 + 46xy + 35y^2\)[/tex], where the coefficients are [tex]\(a = 15\)[/tex], [tex]\(b = 46\)[/tex], and [tex]\(c = 35\)[/tex].
2. Multiply the first and last coefficients:
Multiply [tex]\(a \times c\)[/tex]:
[tex]\(15 \times 35 = 525\)[/tex].
3. Find two numbers that multiply to 525 and add up to 46:
We need two numbers that multiply to give 525 and add up to 46. Those numbers are 25 and 21, because:
[tex]\(25 \times 21 = 525\)[/tex]
[tex]\(25 + 21 = 46\)[/tex].
4. Rewrite the middle term using these two numbers:
Rewrite [tex]\(46xy\)[/tex] as [tex]\(25xy + 21xy\)[/tex]. This changes our trinomial to:
[tex]\(15x^2 + 25xy + 21xy + 35y^2\)[/tex].
5. Group the terms:
Group the terms into two pairs:
[tex]\((15x^2 + 25xy) + (21xy + 35y^2)\)[/tex].
6. Factor out the greatest common factor (GCF) from each group:
- In the first group [tex]\((15x^2 + 25xy)\)[/tex], factor out [tex]\(5x\)[/tex]:
[tex]\(5x(3x + 5y)\)[/tex].
- In the second group [tex]\((21xy + 35y^2)\)[/tex], factor out [tex]\(7y\)[/tex]:
[tex]\(7y(3x + 5y)\)[/tex].
7. Factor out the common binomial factor:
Both groups contain the common factor [tex]\((3x + 5y)\)[/tex].
So we can factor it out:
[tex]\((3x + 5y)(5x + 7y)\)[/tex].
Thus, the trinomial [tex]\(15x^2 + 46xy + 35y^2\)[/tex] factors completely into [tex]\((3x + 5y)(5x + 7y)\)[/tex].
1. Identify the trinomial:
We have [tex]\(15x^2 + 46xy + 35y^2\)[/tex], where the coefficients are [tex]\(a = 15\)[/tex], [tex]\(b = 46\)[/tex], and [tex]\(c = 35\)[/tex].
2. Multiply the first and last coefficients:
Multiply [tex]\(a \times c\)[/tex]:
[tex]\(15 \times 35 = 525\)[/tex].
3. Find two numbers that multiply to 525 and add up to 46:
We need two numbers that multiply to give 525 and add up to 46. Those numbers are 25 and 21, because:
[tex]\(25 \times 21 = 525\)[/tex]
[tex]\(25 + 21 = 46\)[/tex].
4. Rewrite the middle term using these two numbers:
Rewrite [tex]\(46xy\)[/tex] as [tex]\(25xy + 21xy\)[/tex]. This changes our trinomial to:
[tex]\(15x^2 + 25xy + 21xy + 35y^2\)[/tex].
5. Group the terms:
Group the terms into two pairs:
[tex]\((15x^2 + 25xy) + (21xy + 35y^2)\)[/tex].
6. Factor out the greatest common factor (GCF) from each group:
- In the first group [tex]\((15x^2 + 25xy)\)[/tex], factor out [tex]\(5x\)[/tex]:
[tex]\(5x(3x + 5y)\)[/tex].
- In the second group [tex]\((21xy + 35y^2)\)[/tex], factor out [tex]\(7y\)[/tex]:
[tex]\(7y(3x + 5y)\)[/tex].
7. Factor out the common binomial factor:
Both groups contain the common factor [tex]\((3x + 5y)\)[/tex].
So we can factor it out:
[tex]\((3x + 5y)(5x + 7y)\)[/tex].
Thus, the trinomial [tex]\(15x^2 + 46xy + 35y^2\)[/tex] factors completely into [tex]\((3x + 5y)(5x + 7y)\)[/tex].
