Ontario · MPM2D · Grade 10
Ontario Grade 10 Polynomials & Factoring Practice Questions
Expanding in reverse. Pull out the GCF first, then spot the pattern: simple trinomial (a=1), complex (a≠1), difference of squares, or perfect square. Area questions show why it matters.
What you need to know
- GCF first — always check before anything else.
- x²+bx+c — two numbers that multiply to c and add to b.
- ax²+bx+c (a≠1) — numbers that multiply to a·c and add to b, then split and group.
- Special cases — a²−b²=(a−b)(a+b); a²±2ab+b²=(a±b)².
Practice Questions (13)
Question 1 · Factoring Area Applications
A rectangular garden has an area represented by the expression $$A = x^2 + 11x + 28$$ square metres. Determine expressions for the length and width of the garden. If $$x = 5$$ m, find the actual dimensions and the area of the garden.
Show Solution
Let the length and width of the garden be l and w.
Formula/model:
A = x² + 11x + 28
Solve:
Factor the trinomial: A = (x + 7)(x + 4), so the dimensions are (x + 7) and (x + 4).
Substitute x = 5: length = 12 m, width = 9 m, and area = 12 × 9 = 108.
Therefore, dimensions=(x+4) by (x+7), length=12 m, width=9 m, area=108 m^2
Answer: dimensions=(x+4) by (x+7), length=12 m, width=9 m, area=108 m^2
Question 2 · Factoring Area Applications
A rectangular metal plate has outer dimensions of $$(2x + 5)$$ cm and $$(3x + 4)$$ cm. A small rectangular cutout of dimensions $$(x + 1)$$ cm by $$(x + 2)$$ cm is removed. Find a simplified expression for the area of the remaining metal plate.
Show Solution
Let A_outer be the area of the outer rectangle and A_inner be the area of the cutout.
Formula/model:
A_shaded = (2x + 5)(3x + 4) - (x + 1)(x + 2)
Solve:
Expand the areas: A_outer = 6x² + 23x + 20 and A_inner = x² + 3x + 2.
Subtract to find the remaining area: A_shaded = (6x² + 23x + 20) - (x² + 3x + 2) = 5x² + 20x + 18.
Therefore, 5x^2+20x+18
Answer: 5x^2+20x+18
Question 3 · Factoring Area Applications
A picture frame has an outer area represented by $$A = 4x^2 + 20x + 25$$ square centimetres. If the frame is square, find an expression for the side length. If $$x = 8$$ cm, find the perimeter of the outer frame.
Show Solution
Let s be the side length of the square frame, and P be the perimeter.
Formula/model:
s = √(4x² + 20x + 25)
P = 4s
Solve:
Factor the area perfect square trinomial: 4x² + 20x + 25 = (2x + 5)², so side length s = 2x + 5.
Substitute x = 8: s = 2(8) + 5 = 21.
Calculate the perimeter: P = 4 × 21 = 84.
Therefore, side=2x+5, perimeter=84 cm
Answer: side=2x+5, perimeter=84 cm
Question 4 · Complex Trinomials (a ≠ 1)
Factor fully: $$6x^2 + 15x - 9$$
Show Solution
Always check for a GCF first. The GCF of 6x², 15x, and -9 is 3.
Factor out 3:
3(2x² + 5x - 3)
Now, decompose the trinomial inside. Product = 2 × (-3) = -6, sum = 5.
Two numbers: 6 and -1.
3(2x² + 6x - x - 3)
= 3[2x(x + 3) - 1(x + 3)]
= 3(2x - 1)(x + 3).
Answer: 3(2x-1)(x+3)
Question 5 · Factoring Concepts
Which of the following binomials is a difference of squares that can be factored over the integers?
Show Solution
A difference of squares must be in the form a² - b² where both terms are perfect squares separated by subtraction.
4x² - 25 = (2x)² - 5² is a difference of squares.
x² + 9 and 9x² + 16 are sums of squares (not factorable over integers).
x² - 7 is not factorable over integers because 7 is not a perfect square. The correct answer is c.
Answer: $$4x^2 - 25$$
Question 6 · Factoring Concepts
Which of the following trinomials is a perfect square trinomial?
Show Solution
A perfect square trinomial is of the form a² + 2ab + b² = (a+b)².
For x² + 6x + 9: a = x, b = 3. The middle term is 2ab = 2(x)(3) = 6x, and the last term is b² = 3² = 9.
Thus, x² + 6x + 9 = (x + 3)², which is a perfect square trinomial. The correct answer is a.
Answer: $$x^2 + 6x + 9$$
Question 7 · Factoring Concepts
Which of the following trinomials cannot be factored over the integers?
Show Solution
To factor x² + 2x + 5, we need two integers that multiply to 5 and add to 2. The only integer factors of 5 are 1 and 5 (sum = 6) or -1 and -5 (sum = -6). No integers add up to 2. Thus, it is not factorable over integers. The correct answer is c.
Answer: $$x^2 + 2x + 5$$
Question 8 · Common Factoring (GCF)
Factor the greatest common factor (GCF) out of the trinomial: $$15a^3b^2 - 25a^2b^3 + 10a^2b^2$$
Show Solution
Identify the GCF of the three terms:
For coefficients 15, -25, and 10, the GCF is 5.
For variable a (a³, a², a²), the lowest power is a².
For variable b (b², b³, b²), the lowest power is b².
Thus, GCF = 5a²b².
Divide each term by 5a²b²:
15a³b² / 5a²b² = 3a
-25a²b³ / 5a²b² = -5b
10a²b² / 5a²b² = 2
Write as a product: 5a²b²(3a - 5b + 2).
Answer: 5a^2b^2(3a-5b+2)
Question 9 · Common Factoring (GCF)
Factor out the common binomial factor: $$x(y - 4) - (y - 4)$$
Show Solution
In the expression x(y - 4) - (y - 4), there is an implicit coefficient of 1 in front of the second term: x(y - 4) - 1(y - 4).
The common binomial factor is (y - 4).
Factor out (y - 4):
Dividing the first term by (y - 4) leaves x.
Dividing the second term by (y - 4) leaves -1.
Write as a product: (y - 4)(x - 1).
Answer: (y-4)(x-1)
Question 10 · Common Factoring (GCF)
Factor fully by extracting a negative greatest common factor (GCF): $$-2x^2 + 8x$$
Show Solution
Identify the negative GCF of the terms -2x² and 8x:
Since the leading coefficient is negative, we factor out a negative coefficient GCF.
The GCF of -2 and 8 is -2 (taking the negative).
The GCF of x² and x is x.
Thus, GCF = -2x.
Divide each term by -2x:
-2x² / -2x = x
8x / -2x = -4
Write as a product: -2x(x - 4).
Answer: -2x(x-4)
Question 11 · Difference of Squares & Perfect Squares
Factor fully: $$x^2 + 100$$
Show Solution
The binomial x² + 100 is a sum of squares. A sum of squares a² + b² cannot be factored over the integers (it requires imaginary numbers, which are not used in grade 10 math). Therefore, it is not factorable over integers.
Answer: not factorable over integers
Question 12 · Difference of Squares & Perfect Squares
Factor the perfect square trinomial: $$x^2 - 12x + 36$$
Show Solution
Check for a perfect square trinomial: a² - 2ab + b² = (a - b)².
The first term is x² → a = x.
The last term is 36 = 6² → b = 6.
Since the middle term is negative (-12x), use the subtraction identity.
Verify: -2(x)(6) = -12x, which matches.
Factored form: (x - 6)².
Answer: (x-6)^2
Question 13 · Difference of Squares & Perfect Squares
Factor fully: $$25x^2 - 70x + 49$$
Show Solution
Check for a perfect square trinomial: a² - 2ab + b² = (a - b)².
First term: 25x² = (5x)² → a = 5x.
Last term: 49 = 7² → b = 7.
Middle term is negative: -2ab = -2(5x)(7) = -70x, which matches.
Factored form: (5x - 7)².
Answer: (5x-7)^2
Common mistakes
- Missing the GCF and trying to factor what's left.
- Sign errors when c is negative.
- Trying to factor a²+b² over integers — it doesn't factor.
- Stopping after partial factoring; check again.