Introduction
Polynomial arithmetic uses the familiar operations of addition, subtraction, and multiplication, but each operation also has a predictable effect on the degree and leading coefficient. Tracking those features helps you check a calculation and understand the shape of the resulting expression.
The key caution is that the highest-degree terms can cancel when polynomials are added or subtracted. Thus, the degree of a sum or difference is not always the larger of the two input degrees. Multiplication behaves more consistently: for two nonzero polynomials, the degrees add and the leading coefficients multiply.
Polynomial Form and Leading Coefficient
A polynomial with real coefficients can be written in standard form, with powers of \(x\) in descending order. For a nonzero polynomial, its degree is the greatest exponent with a nonzero coefficient, and its leading coefficient is the coefficient of that term. For example, \(5x^3-2x+7\) has degree \(3\) and leading coefficient \(5\).
When comparing terms, it is useful to include missing powers with coefficient zero. For instance, \(5x^3-2x+7\) can be viewed as \(5x^3+0x^2-2x+7\). This makes it clear which coefficients combine during addition and subtraction.
Throughout, the polynomials have real coefficients. If an operation produces the zero polynomial, statements about its degree or leading coefficient do not apply. Keeping this case separate avoids treating a missing leading term as though it had a nonzero coefficient.
Adding and Subtracting Polynomials
To add or subtract polynomials, combine the coefficients of matching powers. Subtraction means subtracting every term of the second polynomial: in \(P(x)-Q(x)\), each coefficient of \(Q\) changes sign before like powers are combined.
If the degrees of two nonzero polynomials differ, the higher-degree term cannot be cancelled by any term of the other polynomial, because the other polynomial has no term of that power. This determines the degree of the result. The leading coefficient depends on whether the higher-degree polynomial is added or subtracted.
If the degrees are equal, the leading terms have the same power, so their coefficients combine. For a sum, those leading coefficients are added; for a difference, the leading coefficient of \(Q\) is subtracted from the leading coefficient of \(P\). If the resulting coefficient is nonzero, it is the leading coefficient of the result. If it is zero, the degree drops and must be found from the next nonzero term. The result might even be the zero polynomial.
Worked Example: Adding and Subtracting Different-Degree Polynomials
Worked Example: Adding and Subtracting Different-Degree Polynomials
Let \(P(x)=4x^4-3x^2+2x-5\) and \(Q(x)=-2x^3+5x^2-7x+1\). Find \(P+Q\) and \(P-Q\), and state each result's degree and leading coefficient.
First, write in the missing \(x^4\) term of \(Q\) with coefficient zero, so the matching powers are visible:
Add matching coefficients:
The degree is \(4\), and the leading coefficient is \(4\). For the subtraction, subtract each coefficient of \(Q\):
This result also has degree \(4\) and leading coefficient \(4\). As a check, at \(x=1\), \(P(1)=-2\) and \(Q(1)=-3\). The calculated sum gives \(-5=P(1)+Q(1)\), and the calculated difference gives \(1=P(1)-Q(1)\).
Why the Addition and Subtraction Rules Hold
Suppose \(P\) has degree \(m\) and \(Q\) has degree \(n\), with \(m>n\). Write the leading term of \(P\) as \(a_mx^m\), where \(a_m\ne0\). Since \(Q\) has degree \(n<m\), it contains no \(x^m\) term. Therefore, in both \(P+Q\) and \(P-Q\), the coefficient of \(x^m\) remains \(a_m\), and every other term has a lower power. The result has degree \(m\) and leading coefficient \(a_m\).
Now suppose \(n>m\), and write the leading term of \(Q\) as \(b_nx^n\), where \(b_n\ne0\). The polynomial \(P\) has no \(x^n\) term. Thus, \(P+Q\) has leading term \(b_nx^n\), whereas \(P-Q\) has leading term \(-b_nx^n\), because the entire polynomial \(Q\) is subtracted. This proves both cases in the different-degree rule.
If the degrees are equal, say \(m=n\), the coefficient of the shared highest power \(x^n\) is \(a_n+b_n\) in \(P+Q\) and \(a_n-b_n\) in \(P-Q\). If that coefficient is nonzero, the degree remains \(n\). If it is zero, the \(x^n\) terms cancel, so lower powers must be checked. This explains why equal degrees alone do not determine the degree of the sum or difference.
Worked Example: Equal-Degree Terms Cancel
Worked Example: Equal-Degree Terms Cancel
Let \(P(x)=3x^3+2x-4\) and \(Q(x)=3x^3-x^2+5x+1\). Find \(P-Q\), then determine its degree and leading coefficient.
Both polynomials have degree \(3\) and leading coefficient \(3\). Subtract coefficient by coefficient, including the missing \(x^2\) term in \(P\):
The \(x^3\) terms cancel. The highest remaining nonzero power is \(x^2\), so the result has degree \(2\) and leading coefficient \(1\). Check at \(x=2\): \(P(2)=24\) and \(Q(2)=31\), giving \(P(2)-Q(2)=-7\). The calculated result gives \(2^2-3(2)-5=4-6-5=-7\), as required.
Multiplying Polynomials
To multiply polynomials, distribute each term of one polynomial across every term of the other, then combine like powers. When multiplying individual terms, multiply their coefficients and add their exponents: \( (ax^r)(bx^s)=abx^{r+s}\).
For two nonzero polynomials, the product has degree equal to the sum of their degrees, and its leading coefficient is the product of their leading coefficients. Unlike addition and subtraction, the highest-degree product term cannot be cancelled by another pair of terms: only the two leading terms can produce the sum of the two highest degrees.
Deriving the Multiplication Rule
Write \(P(x)=a_mx^m+\) terms of lower degree and \(Q(x)=b_nx^n+\) terms of lower degree, with \(a_m\ne0\) and \(b_n\ne0\). Multiplying the leading terms gives \(a_mb_nx^{m+n}\). Because all other terms in \(P\) have powers below \(m\), and all other terms in \(Q\) have powers below \(n\), every other pair of terms has a product whose exponent is less than \(m+n\). Therefore, no other product contributes to the power \(x^{m+n}\). Its coefficient is \(a_mb_n\ne0\), so the product has degree \(m+n\) and that leading coefficient.
Worked Example: Multiplying and Tracking the Leading Term
Worked Example: Multiplying and Tracking the Leading Term
Multiply \((2x^2-3x+1)(x^2+4x-2)\), and identify the degree and leading coefficient.
Distribute each term of the first factor across the second:
The degrees add to \(2+2=4\), and the leading coefficients multiply to \(2\cdot1=2\). Thus the result has degree \(4\) and leading coefficient \(2\). Check by evaluating at \(x=2\): the factors give \(2(2^2)-3(2)+1=3\) and \(2^2+4(2)-2=10\), so their product is \(30\). The expanded result gives \(2(2^4)+5(2^3)-15(2^2)+10(2)-2=32+40-60+20-2=30\).
A Common Mistake: Forgetting the Sign in Subtraction
A frequent error is to copy the leading coefficient of the higher-degree polynomial without accounting for whether that polynomial is being subtracted. For example, let \(P(x)=1\) and \(Q(x)=2x\). Since \(Q\) has higher degree, it determines the degree of \(P-Q\), but its leading coefficient changes sign:
The result has degree \(1\) and leading coefficient \(-2\). The incorrect calculation \(P-Q=2x+1\) treats the leading term of \(Q\) as positive even though \(Q\) is subtracted. Check at \(x=1\): the original expression gives \(1-2=-1\), and the corrected polynomial gives \(-2+1=-1\); the incorrect expression would give \(3\).
Why These Rules Matter
Degree and leading coefficient provide quick checks on polynomial calculations. Before expanding a product, you can predict its degree and leading coefficient. After adding or subtracting, you can check whether the highest terms should remain or cancel. These predictions do not replace the calculation, but they can reveal a sign error or an incorrectly combined term.
There are two limits to keep in mind. First, when equal-degree terms cancel, inspect lower powers rather than assuming the original degree remains. Second, a zero result has no leading coefficient and, under the convention used here, no degree. For instance, \(P-P=0\) for every polynomial \(P\); it would be incorrect to assign that result the degree of \(P\).
Summary
For addition and subtraction, combine coefficients of like powers. If two nonzero input polynomials have different degrees, their sum and difference have the larger degree. In subtraction, the leading coefficient is negated when the higher-degree polynomial is the one being subtracted. If the input degrees are equal, combine their leading coefficients; cancellation may lower the degree or produce zero.
For multiplication of nonzero polynomials, add the degrees and multiply the leading coefficients. The leading product term comes only from multiplying the two leading terms. These rules give both a way to calculate and a way to check polynomial arithmetic.
Check Your Understanding
Answer each question. For calculations, state the degree and leading coefficient whenever the result is nonzero.
- If \(P\) has degree \(5\) and leading coefficient \(3\), and \(Q\) has degree \(2\), what are the degree and leading coefficient of \(P+Q\)? What are they for \(P-Q\)?
- Let \(A(x)=4x^3+x\) and \(B(x)=4x^3-2x^2+3\). Find \(A-B\). Why is its degree less than \(3\)?
- Can the sum of two degree-\(4\) polynomials have degree less than \(4\)? Explain what must happen to their leading coefficients.
- If nonzero polynomials \(R\) and \(S\) have degrees \(3\) and \(6\), and leading coefficients \(-2\) and \(5\), respectively, what are the degree and leading coefficient of \(RS\)?
- What is the degree of the zero polynomial under the convention in this tutorial, and why is it different from the degree of a nonzero polynomial?
