Algebraic expressions · Ages 13–14
Polynomials & identities worksheet
Covers: What a polynomial is, Adding and subtracting polynomials, A monomial times a polynomial, Multiplying polynomials, Identities and equations, Comparing coefficients, Expanding perfect squares, Expanding the difference of two squares, The sum and difference of two cubes
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Polynomials & identities worksheet
Algebraic expressions · Ages 13–14
Numsprite
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Name:Date:Score:／48
Write each answer in the space. For word problems, show your working and write a sentence answer.
Part A: Working out and filling in (2 marks each)
1.In the polynomial 7x3 + 5x2 − x + 15, the coefficient of x2 is , the coefficient of x is and the constant term is .
Fill in each blank
Answer:
2.(8x + 3) + (5x + 14) = x +
Fill in each blank
Answer:
3.After simplifying 9x(3x + 6) − 24x2, the coefficient of x2 is .
Answer:
4.(x + 4)(x + 6) = x2 + x +
Fill in each blank
Answer:
5.(x + 8)2 = x2 + x +
Fill in each blank
Answer:
6.Work it out quickly with the difference of two cubes: 123 − 33 = (12 − 3) ×
Answer:
Part B: Choose, or true or false (2 marks each)
7.The constant term of the polynomial 6x + 16 + 2x2 is 16.
True or false?
Answer:
8.4x4y3 and 3x4y3 are like terms.
True or false?
Answer:
9.−4x(9x − 4) = ?
- A. −36x2 + 4
- B. −36x2 − 16x
- C. 36x2 − 16x
- D. −36x2 + 16x
Choose the right answer
Answer:
10.Which of these is not an identity?
- A. (x − 6)2 = x2 − 36
- B. 3(x − 7) + 18 = 3x − 3
- C. (x + 6)2 = x2 + 12x + 36
- D. (x − 7)(x + 7) = x2 − 49
Choose the right answer
Answer:
11.If (px + 8)(x + 2) ≡ 3x2 + qx + 16, what are p and q?
- A. p = 3, q = 26
- B. p = 3, q = 10
- C. p = 3, q = 14
- D. p = 16, q = 14
Choose the right answer
Answer:
12.(5x + 6)(5x − 6) = ?
- A. 25x2 − 60x − 36
- B. 25x2 + 36
- C. 25x2 − 36
- D. 5x2 − 36
Choose the right answer
Answer:
Part C: Word problems (4 marks each)
13.Ming wants to write the polynomial 9x − 6x3 + 15 − x2 in ascending powers of x. Which is right?
- A. 15 + 9x − x2 − 6x3
- B. 15 + 9x − 6x3 − x2
- C. −6x3 − x2 + 9x + 15
- D. 15 + 9x + x2 + 6x3
Choose the right answer
Working:
Answer:
14.A rectangle is (5x + 1) m long and (2x − 6) m wide. Its length is (x + ) m more than its width.
Fill in each blank
Working:
Answer:
15.Prove (x + 10)2 − (x − 10)2 ≡ 40x. Left side = (x2 + x + 100) − (x2 − x + 100) = x = right side.
Fill in each blank
Working:
Answer:
16.If (x + p)(x + q) ≡ x2 − x − 30 and p > q, then p = and q = .
Fill in each blank
Working:
Answer:
17.Ming says: 56 × 64 = 602 + 42.
True or false?
Working:
Answer:
18.A cube with edges x cm long and a cube with edges 2 cm long have a total volume of (x3 + 8) cm3 = (x + 2)(x2 − 2x + 4) cm3. When x = 10, the second factor is and the total volume is cm3.
Fill in each blank
Working:
Answer:
Show the answers and working
Polynomials & identities worksheet: Answers and working
Part A: Working out and filling in
1. 5, −1, 15
Each sign goes with the term after it: x2's coefficient is 5; x alone has coefficient 1 and −x has −1, so here it is −1; the constant is 15.
2. 13, 17
Combine like terms: 8x + 5x = 13x and 3 + 14 = 17.
3. 3
9x × 3x = 27x2, then take away 24x2: 27 − 24 = 3.
4. 10, 24
Every term times every term: x2 + 6x + 4x + 24, which combines to x2 + 10x + 24.
5. 16, 64
First squared, x2; twice the product, 2 × x × 8 = 16x; last squared, 82 = 64.
6. 189
a3 − b3 ≡ (a − b)(a2 + ab + b2): the second factor is 122 + 12 × 3 + 32 = 189.
Part B: Choose, or true or false
7. True
The constant term is the one with no letter: 16. 6 is the coefficient of x.
8. True
Like terms have the same letters, each to the same index: here x's indices are 4 and 4.
9. D. −36x2 + 16x
−4x times each term: (−4x) × 9x = −36x2 and (−4x) × (−4) = +16x.
10. A. (x − 6)2 = x2 − 36
(x − 6)2 = (x − 6)(x − 6) = x2 − 12x + 36, not x2 − 36. Putting in x = 0 shows it too: 36 on the left, −36 on the right.
11. C. p = 3, q = 14
The left is px2 + (p × 2 + 8)x + 16. Comparing x2's coefficients, p = 3; comparing x's, q = 3 × 2 + 8 = 14.
12. C. 25x2 − 36
(a + b)(a − b) ≡ a2 − b2: (5x)2 − 62 = 25x2 − 36.
Part C: Word problems
Marking: 2 for the working, 1 for the answer, 1 for the sentence.
13. A. 15 + 9x − x2 − 6x3
The degrees: 15 is 0, 9x is 1, −x2 is 2, −6x3 is 3. Lowest first, each sign with its term.
14. 3, 7
Length − width = (5x + 1) − (2x − 6) = 5x + 1 − 2x + 6 = 3x + 7. Taking away −6 adds 6.
15. 20, 20, 40
Expanding both squares, x2 and 100 cancel, leaving 20x + 20x = 40x, the right side: so it is an identity.
16. 5, −6
The left is x2 + (p + q)x + pq. Comparing: p + q = −1 and pq = −30. The two numbers with sum −1 and product −30 are 5 and −6.
17. False
56 × 64 = (60 − 4)(60 + 4), a sum times a difference: 602 − 42.
18. 84, 1008
102 − 2 × 10 + 4 = 84; the total volume = (10 + 2) × 84 = 1008, the same as 103 + 8.
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