To expand the expression \((4x^2 + x + 5)(x^2 - x + 1)\), we can apply the distributive property (also known as the FOIL method for binomials). We'll multiply each term in the first polynomial by each term in the second polynomial.
-
Multiply \(4x^2\) by each term in \((x^2 - x + 1)\):
- \(4x^2 \cdot x^2 = 4x^4\)
- \(4x^2 \cdot (-x) = -4x^3\)
- \(4x^2 \cdot 1 = 4x^2\)
-
Multiply \(x\) by each term in \((x^2 - x + 1)\):
- \(x \cdot x^2 = x^3\)
- \(x \cdot (-x) = -x^2\)
- \(x \cdot 1 = x\)
-
Multiply \(5\) by each term in \((x^2 - x + 1)\):
- \(5 \cdot x^2 = 5x^2\)
- \(5 \cdot (-x) = -5x\)
- \(5 \cdot 1 = 5\)
Now, we can combine all the terms obtained from the multiplications:
Gather the terms:
-
From \(4x^2\):
- \(4x^4 - 4x^3 + 4x^2\)
-
From \(x\):
- \(x^3 - x^2 + x\)
-
From \(5\):
- \(5x^2 - 5x + 5\)
Combine like terms:
- Overall expression:
\(4x^4\)
- Combine the \(x^3\) terms: \(-4x^3 + x^3 = -3x^3\)
- Combine the \(x^2\) terms: \(4x^2 - x^2 + 5x^2 = 8x^2\)
- Combine the \(x\) terms: \(x - 5x = -4x\)
- The constant term is \(5\).
Final Combined Expression:
Putting it all together, we get: \[ 4x^4 - 3x^3 + 8x^2 - 4x + 5 \]
Thus, the expanded expression in standard form is: \[ \boxed{4x^4 - 3x^3 + 8x^2 - 4x + 5} \]
1 answer