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Add, subtract or multiply polynomials

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Operate on polynomials

Add and subtract polynomials by matching equal powers. Multiply polynomials by distributing every term, adding exponents for the same variable, and then collecting like powers

\[\left(ax^m\right)\left(bx^n\right)=abx^{m+n}\]
  1. 1Align equal powers When adding or subtracting, combine coefficients only for terms with the same power \(\displaystyle 2x^2+5x^2=7x^2\)
  2. 2Distribute subtraction A negative sign outside a bracket changes the sign of every term inside it \(\displaystyle -\left(3x-4\right)=-3x+4\)
  3. 3Distribute every product Multiply every term in one polynomial by every term in the other polynomial \(\displaystyle \left(a+b\right)\left(c+d\right)=ac+ad+bc+bd\)
  4. 4Add exponents in products When powers of the same variable are multiplied, add their exponents \(\displaystyle \left(2x^2\right)\left(3x^3\right)=6x^5\)
Year 11, VCE Units 1 & 2, Chapter 4, 4 difficulty levels, No calculatorBuild a worksheet

Polynomial operations

Add and subtract polynomials by matching equal powers. Multiply polynomials by distributing every term, adding exponents for the same variable, and then collecting like powers

\[\left(ax^m\right)\left(bx^n\right)=abx^{m+n}\]
  1. Align equal powers When adding or subtracting, combine coefficients only for terms with the same power \(\displaystyle 2x^2+5x^2=7x^2\)
  2. Distribute subtraction A negative sign outside a bracket changes the sign of every term inside it \(\displaystyle -\left(3x-4\right)=-3x+4\)
  3. Distribute every product Multiply every term in one polynomial by every term in the other polynomial \(\displaystyle \left(a+b\right)\left(c+d\right)=ac+ad+bc+bd\)
  4. Add exponents in products When powers of the same variable are multiplied, add their exponents \(\displaystyle \left(2x^2\right)\left(3x^3\right)=6x^5\)

Worked example

Expand and simplify the polynomial product below

\[\left(3x+2\right)\left(x-5\right)\]
  1. \(\displaystyle 3x\times x=3x^2,\qquad3x\times(-5)=-15x\)Multiply the first term in the first bracket by both terms in the second bracket
  2. \(\displaystyle 2\times x=2x,\qquad2\times(-5)=-10\)Multiply the second term in the first bracket by both terms in the second bracket
  3. \(\displaystyle 3x^2-15x+2x-10=3x^2-13x-10\)Collect the two like x terms to obtain the expanded polynomial

Common mistakes

Combining unlike powers
✕\(\displaystyle 4x^3-2x=2x^3\)✓\(\displaystyle 4x^3-2x\ne2x^3\)
Terms with different powers cannot be combined by adding their coefficients
Not distributing a subtraction
✕\(\displaystyle -\left(3x-4\right)=-3x-4\)✓\(\displaystyle -\left(3x-4\right)=-3x+4\)
The negative sign changes the sign of every term inside the bracket
Using the wrong product exponent
✕\(\displaystyle \left(2x^2\right)\left(3x^3\right)=6x^6\)✓\(\displaystyle \left(2x^2\right)\left(3x^3\right)=6x^5\)
The coefficients multiply, while the exponents of the same variable are added

Difficulty levels

introTwo small polynomials are added by aligning equal powers
standardSubtraction and scaled combinations include missing powers
challengeProducts and deliberate leading-term cancellation require complete distribution
examA diagnostic result choice is followed by coefficient entries and a degree check

Questions about this topic

Which polynomial terms are like terms?+

They have the same variable raised to the same power. Only like terms can be combined by adding or subtracting coefficients

What does distribution mean when multiplying polynomials?+

Each term in one polynomial is multiplied by each term in the other polynomial, and the resulting terms are collected

Why are exponents added when powers are multiplied?+

Multiplying powers joins their repeated factors, so the exponents are added when powers of the same variable are multiplied

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, Mathematical Methods, chapter 4: Cubic polynomials

In this chapter

Recognise and classify a polynomialEvaluate a polynomial or recover a coefficient parameterAdd, subtract or multiply polynomialsDivide polynomials and state quotient and remainderAnalyse a cubic graph or determine its ruleDetermine parameters in a polynomial identityApply the remainder and factor theoremsMove between roots, factors, polynomial rules and equationsSolve a cubic inequalityIntersections of a cubic with a line or quadraticFormulate and interpret a cubic model