Expand products systematically and use coefficient comparisons.
Notes and key methods
Multiply every term in one factor by every term in the other, then collect equal powers. A missing power still has a zero coefficient when performing division or comparing two polynomial expressions.
An identity holds for all x; its coefficients can therefore be compared. An equation holds only at its solutions. Substitution into an expanded polynomial provides a useful independent arithmetic check.
Check the conditions
Retain signs inside every bracket.
Align the same powers before adding coefficients.
Three worked examples
Example 1 · Polynomial arithmetic and coefficients
Let \(\displaystyle P(x)=\)\(\left(x - 5\right) \left(x - 1\right) \left(x + 5\right)\) and \(\displaystyle u=4\).
Expand P(x) completely.
Find the coefficient of x in P.
Find the constant term in P.
Find P(u).
Show the reasoning
(a)
\(x^{2} - 6 x + 5\) multiplied by \(x + 5\) gives \(x^{3} - x^{2} - 25 x + 25\).
Answer: \(x^{3} - x^{2} - 25 x + 25\)
(b)
Collect all degree-one products.
\(x^{3} - x^{2} - 25 x + 25\)
Answer: \(-25\).
(c)
Multiply the three constant terms.
\(25\)
Answer: \(25\).
(d)
Use the factored form to check the arithmetic.
\(-27\)
Answer: \(-27\).
Example 2 · Polynomial arithmetic and coefficients
Let \(\displaystyle P(x)=\)\(\left(x - 6\right) \left(x - 2\right) \left(x + 2\right)\) and \(\displaystyle u=5\).
Expand P(x) completely.
Find the coefficient of x in P.
Find the constant term in P.
Find P(u).
Show the reasoning
(a)
\(x^{2} - 8 x + 12\) multiplied by \(x + 2\) gives \(x^{3} - 6 x^{2} - 4 x + 24\).
Answer: \(x^{3} - 6 x^{2} - 4 x + 24\)
(b)
Collect all degree-one products.
\(x^{3} - 6 x^{2} - 4 x + 24\)
Answer: \(-4\).
(c)
Multiply the three constant terms.
\(24\)
Answer: \(24\).
(d)
Use the factored form to check the arithmetic.
\(-21\)
Answer: \(-21\).
Example 3 · Polynomial arithmetic and coefficients
Let \(\displaystyle P(x)=\)\(\left(x - 7\right) \left(x - 3\right) \left(x + 3\right)\) and \(\displaystyle u=1\).
Expand P(x) completely.
Find the coefficient of x in P.
Find the constant term in P.
Find P(u).
Show the reasoning
(a)
\(x^{2} - 10 x + 21\) multiplied by \(x + 3\) gives \(x^{3} - 7 x^{2} - 9 x + 63\).
Answer: \(x^{3} - 7 x^{2} - 9 x + 63\)
(b)
Collect all degree-one products.
\(x^{3} - 7 x^{2} - 9 x + 63\)
Answer: \(-9\).
(c)
Multiply the three constant terms.
\(63\)
Answer: \(63\).
(d)
Use the factored form to check the arithmetic.
\(48\)
Answer: \(48\).
Common mistakes
Do not multiply only matching powers.
A polynomial identity is stronger than one equation at one x.