Algebraic expressions · Ages 13–14
Polynomials & identities
From “What a polynomial is” to “The sum and difference of two cubes”, step by step, each skill with its methods, worked examples and answers.
Skills and methods in this topic, easy to hard
- What a polynomial isA sign goes with the term after it; the degree is the highest term's
Method4x3 − 2x + 7: three terms, x's coefficient −2, degree 3
Another way: Rewrite it all as adding4x3 + (−2x) + 7
- Adding and subtracting polynomialsTaking a bracket away changes every sign inside it
Method(3x2 − 2x + 5) − (x2 − 4x + 1) = 2x2 + 2x + 4
Another way: Line up like terms in columnsx2 under x2, x under x, number under number
- A monomial times a polynomialThe term outside times every term inside; same letters, add the indices
Method3x(2x − 5) = 6x2 − 15x
Another way: A rectangle's area3x × 2x = 6x2, 3x × 5 = 15x
- Multiplying polynomialsEvery term times every term, then combine like terms
Method(x + 3)(2x − 5) = 2x2 + x − 15
Another way: The grid methodCells 2x2, −5x, 6x, −15, combined to 2x2 + x − 15
- Identities and equationsAn identity: both sides simplify to exactly the same
Method2(x + 3) ≡ 2x + 6; 2x + 6 = 10 only when x = 2
Another way: Look for a value that fails(x + 1)2 and x2 + 1 at x = 1: 4 ≠ 2
- Comparing coefficientsExpand, put in the same order, then match term by term
Method(x + 2)(x + k) ≡ x2 + 7x + m: 2 + k = 7, k = 5, m = 10
Another way: Put in handy valuesx = 0: 2k = m
- Expanding perfect squaresFirst squared, twice the product, last squared: keep the middle term
Method(3x − 2)2 = 9x2 − 12x + 4
Another way: As two brackets multiplied(3x − 2)(3x − 2) = 9x2 − 6x − 6x + 4
- Expanding the difference of two squaresSum times difference = difference of squares: the middle cancels
Method(2x + 5)(2x − 5) = 4x2 − 25
Another way: Every term by every term4x2 − 10x + 10x − 25 = 4x2 − 25
- The sum and difference of two cubesFirst bracket the same sign, the middle the opposite, the last added
Method(x + 2)(x2 − 2x + 4) = x3 + 8
Another way: Every term by every termx3 − 2x2 + 4x + 2x2 − 4x + 8 = x3 + 8
Worked examples
From different skills in this topic. Have a go first, then open each one for the answer and the working.
Example 1What a polynomial is →
The degree of the polynomial 9x4 + 7x3y2 − 9 is .
Show the answer and working
Answer: 5
9x4 has degree 4; 7x3y2 has degree 3 + 2 = 5; the constant, 0. A polynomial's degree is its highest term's.
Example 2Identities and equations →
For 5(x − 9) = 5x − k to be an identity, k = .
Show the answer and working
Answer: 45
The left expands to 5x − 45. Both sides must match exactly, so k = 45.
Example 3The sum and difference of two cubes →
Expand (x − 5)(x2 + 5x + 25). What do you get?
- A. x3 − 15x2 + 75x − 125
- B. x3 + 125
- C. x3 − 25
- D. x3 − 125
Choose the right answer
Show the answer and working
Answer: D. x3 − 125
The difference of cubes: (x − 5)(x2 + 5x + 25) = x3 − 53 = x3 − 125.
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistakeAll secondary maths topics →