Expand the expression to a polynomial in standard form: left bracket, 4, x, squared, plus, 5, x, plus, 3, right bracket, left bracket, 2, x, squared, minus, 2, x, plus, 1, right bracket (4x 2 +5x+3)(2x 2 −2x+1)
The Correct Answer and Explanation is:
To expand the expression (4×2+5x+3)(2×2−2x+1)(4x^2 + 5x + 3)(2x^2 – 2x + 1)(4×2+5x+3)(2×2−2x+1), we will use the distributive property, also known as the FOIL method (First, Outer, Inner, Last), and carefully multiply each term from the first binomial by every term in the second binomial.
Step-by-step expansion:
- Multiply the first term in the first binomial by all terms in the second binomial: (4×2)×(2×2−2x+1)=4×2(2×2)+4×2(−2x)+4×2(1)(4x^2) \times (2x^2 – 2x + 1) = 4x^2(2x^2) + 4x^2(-2x) + 4x^2(1)(4×2)×(2×2−2x+1)=4×2(2×2)+4×2(−2x)+4×2(1) This simplifies to: 8×4−8×3+4x28x^4 – 8x^3 + 4x^28×4−8×3+4×2
- Multiply the second term in the first binomial by all terms in the second binomial: (5x)×(2×2−2x+1)=5x(2×2)+5x(−2x)+5x(1)(5x) \times (2x^2 – 2x + 1) = 5x(2x^2) + 5x(-2x) + 5x(1)(5x)×(2×2−2x+1)=5x(2×2)+5x(−2x)+5x(1) This simplifies to: 10×3−10×2+5x10x^3 – 10x^2 + 5x10x3−10×2+5x
- Multiply the third term in the first binomial by all terms in the second binomial: (3)×(2×2−2x+1)=3(2×2)+3(−2x)+3(1)(3) \times (2x^2 – 2x + 1) = 3(2x^2) + 3(-2x) + 3(1)(3)×(2×2−2x+1)=3(2×2)+3(−2x)+3(1) This simplifies to: 6×2−6x+36x^2 – 6x + 36×2−6x+3
Combine all the results:
Now, we add all the terms together:8×4−8×3+4×2+10×3−10×2+5x+6×2−6x+38x^4 – 8x^3 + 4x^2 + 10x^3 – 10x^2 + 5x + 6x^2 – 6x + 38×4−8×3+4×2+10×3−10×2+5x+6×2−6x+3
Simplify the expression by combining like terms:
- Combine −8×3-8x^3−8×3 and 10x310x^310×3: −8×3+10×3=2×3-8x^3 + 10x^3 = 2x^3−8×3+10×3=2×3
- Combine 4x24x^24×2, −10×2-10x^2−10×2, and 6x26x^26×2: 4×2−10×2+6×2=0x24x^2 – 10x^2 + 6x^2 = 0x^24×2−10×2+6×2=0x2 (which cancels out)
- Combine 5x5x5x and −6x-6x−6x: 5x−6x=−x5x – 6x = -x5x−6x=−x
- The constant term is just 3.
Final polynomial:
The expanded expression in standard form is:8×4+2×3−x+38x^4 + 2x^3 – x + 38×4+2×3−x+3
Explanation:
We followed the distributive property by multiplying each term in the first binomial by every term in the second binomial. After distributing, we combined like terms and simplified the expression. The final result is a polynomial of degree 4 (the highest power of xxx) with terms in descending order of degree: 8×4+2×3−x+38x^4 + 2x^3 – x + 38×4+2×3−x+3. This is the polynomial in standard form.
