Q. Express as a trinomial: \( (2x + 6)(x + 9) = 2x^2 + 18x + 6x + 54 = 2x^2 + 24x + 54\).

Q: What is the expanded trinomial for (2x+6)(x+9) ?

Expand by FOIL: 2x\cdot x=2x^2,\;2x\cdot9=18x,\;6\cdot x=6x,\;6\cdot9=54. Combine: 2x^2+24x+54..

Q: What does FOIL stand for and how is it used here?

FOIL = First, Outer, Inner, Last. For (2x+6)(x+9) : First 2x\cdot x, Outer 2x\cdot 9, Inner 6\cdot x, Last 6\cdot 9. Sum the four products..

Q: Can I simplify before expanding?

Yes. Factor out common factor: 2x+6=2(x+3). Then (2x+6)(x+9)=2(x+3)(x+9)=2(x^2+12x+27)=2x^2+24x+54.

Q: What are the roots/zeros of the trinomial ?

Solve 2x^2+24x+54=0. Divide by 2: x^2+12x+27=0. Roots: x=-3 and x=-9.

Q: What is the degree and leading coefficient?

Degree is 2 (quadratic). Leading coefficient is 2 from 2x^2+24x+54.\n

Q: How can I check my expansion is correct?

Substitute convenient x values e.g., x=0 gives 54 or re-multiply using distribution/FOIL and confirm identical results for multiple x values.

Q: What is the vertex of the parabola y=2x^2+24x+54?.

Vertex x-coordinate: -b/(2a)=-24/(4)=-6. y-value: 2(-6)^2+24(-6)+54=-18. Vertex: (-6,-18).

Answer

\[
(2x+6)(x+9)=2x^2+18x+6x+54=2x^2+24x+54.
\]

Detailed Explanation

Problem

Express as a trinomial: \left(2x+6\right)\left(x+9\right)

Factor any common factor in a binomial (optional but helpful):
Notice 2x+6 has a common factor 2. So

\left(2x+6\right)\left(x+9\right)=\left(2\left(x+3\right)\right)\left(x+9\right)
Reorder scalar factors:
Bring the 2 outside so we expand the simpler binomials:

=2\left(x+3\right)\left(x+9\right)
Expand the product of the two binomials using FOIL (First, Outer, Inner, Last):
Compute each product:

x\cdot x = x^{2}

x\cdot 9 = 9x

3\cdot x = 3x

3\cdot 9 = 27

Sum these to get

\left(x+3\right)\left(x+9\right)=x^{2}+9x+3x+27=x^{2}+12x+27
Multiply by the scalar 2:
2\left(x^{2}+12x+27\right)=2x^{2}+24x+54

Final trinomial: 2x^{2}+24x+54

See full solution

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Algebra FAQs

What is the expanded trinomial for \( (2x+6)(x+9) \)?

Expand by FOIL: \(2x\cdot x=2x^2,\;2x\cdot9=18x,\;6\cdot x=6x,\;6\cdot9=54\). Combine: \(2x^2+24x+54\)..

What does FOIL stand for and how is it used here?

FOIL = First, Outer, Inner, Last. For \( (2x+6)(x+9) \): First \(2x\cdot x\), Outer \(2x\cdot 9\), Inner \(6\cdot x\), Last \(6\cdot 9\). Sum the four products..

Can I simplify before expanding?

Yes. Factor out common factor: \(2x+6=2(x+3)\). Then \( (2x+6)(x+9)=2(x+3)(x+9)=2(x^2+12x+27)=2x^2+24x+54\).

What are the roots/zeros of the trinomial ?

Solve \(2x^2+24x+54=0\). Divide by 2: \(x^2+12x+27=0\). Roots: \(x=-3\) and \(x=-9\).

What is the degree and leading coefficient?

Degree is 2 (quadratic). Leading coefficient is 2 from \(2x^2+24x+54\).\n

What is the constant term and how is it obtained?

How can I check my expansion is correct?

Substitute convenient \(x\) values \(e.g., \(x=0\) gives 54\) or re-multiply using distribution/FOIL and confirm identical results for multiple \(x\) values.

What is the vertex of the parabola \(y=2x^2+24x+54\)?.

Vertex x-coordinate: \(-b/(2a)=-24/(4)=-6\). y-value: \(2(-6)^2+24(-6)+54=-18\). Vertex: \((-6,-18)\).

Expand the trinomial to a polynomial.
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