Quick Review Flashcards - Click to flip and test your knowledge!
Question
What is the mathematical definition of factorisation?
Answer
The process of writing an algebraic expression as the product of its factors.
Question
If a polynomial is expressed as the product of two or more expressions, each of those expressions is called a _____ of the polynomial.
Answer
factor
Question
Factorisation is conceptually considered the reverse of which mathematical operation?
Answer
Multiplication.
Question
What are the factors of the polynomial $x^2 + 5x + 6$?
Answer
$(x + 3)$ and $(x + 2)$.
Question
In the 'Taking out common factors' method, what must be found for all terms in the expression?
Answer
The Highest Common Factor (H.C.F.).
Question
What is the H.C.F. of the terms in the expression $6a^2 - 3ax$?
Answer
$3a$.
Question
Factorise the expression $8ab^2 + 12a^2b$.
Answer
$4ab(2b + 3a)$.
Question
Factorise the expression $4(x + y)^2 - 3(x + y)$.
Answer
$(x + y)(4x + 4y - 3)$.
Question
Factorise the expression $x(a - 5) + y(5 - a)$.
Answer
$(a - 5)(x - y)$.
Question
Factorisation by grouping is generally used for expressions containing an _____ number of terms.
Answer
even
Question
What is the first step in the procedure for factorisation by grouping?
Answer
Arrange the terms of the expression into groups such that each group has a common factor.
Question
Factorise the expression $ab^2 - (a - 1)b - 1$ using grouping.
Answer
$(b - 1)(ab + 1)$.
Question
When factorising a trinomial of the form $ax^2 + bx + c$, the middle term $b$ is split into two parts such that their product is equal to _____.
Answer
$ac$
Question
When factorising a trinomial of the form $ax^2 + bx + c$, the middle term $b$ is split into two parts such that their sum is equal to _____.
Answer
$b$
Question
Factorise the trinomial $x^2 - 5x + 6$.
Answer
$(x - 3)(x - 2)$.
Question
Factorise the trinomial $x^2 - 5x - 6$.
Answer
$(x - 6)(x + 1)$.
Question
Factorise the expression $3x^2 - 11x - 4$ by splitting the middle term.
Answer
$(x - 4)(3x + 1)$.
Question
Factorise the expression $7 - 12x - 4x^2$.
Answer
$(1 - 2x)(7 + 2x)$.
Question
What is a trinomial or quadratic expression in terms of real numbers $a, b,$ and $c$?
Answer
An expression of the form $ax^2 + bx + c$, where $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is a constant.
Question
Under what condition involving $b^2 - 4ac$ is the trinomial $ax^2 + bx + c$ factorisable?
Answer
It is factorisable if the value of $b^2 - 4ac$ is a perfect square.
Question
Using the discriminant method, is $5x^2 + 17x + 6$ factorisable?
Answer
Yes, because $b^2 - 4ac = 169$, which is a perfect square.
Question
Why is the trinomial $3x^2 - 8x - 15$ considered not factorisable?
Answer
Because its discriminant $b^2 - 4ac = 244$ is not a perfect square.
Question
State the identity for the difference of two squares: $x^2 - y^2 = $ _____.
Answer
$(x + y)(x - y)$.
Question
Factorise the expression $9(x - y)^2 - (x + 2y)^2$.
Answer
$(4x - y)(2x - 5y)$.
Question
Factorise the expression $48x^3 - 27x$.
Answer
$3x(4x + 3)(4x - 3)$.
Question
Factorise the expression $16a^4 - b^4$ completely.
Answer
$(4a^2 + b^2)(2a + b)(2a - b)$.
Question
Factorise the expression $16x^2 - y^2 + 4yz - 4z^2$.
Answer
$(4x + y - 2z)(4x - y + 2z)$.
Question
Factorise the expression $(1 - x^2)(1 - y^2) + 4xy$.
Answer
$(1 + xy + x - y)(1 + xy - x + y)$.
Question
Factorise the expression $x^4 + x^2y^2 + y^4$.
Answer
$(x^2 + y^2 + xy)(x^2 + y^2 - xy)$.
Question
Factorise the expression $(p^2 + q^2 - r^2)^2 - 4p^2q^2$.
Answer
$(p + q + r)(p + q - r)(p - q + r)(p - q - r)$.
Question
Formula: $a^3 + b^3$
Answer
Definition: $(a + b)(a^2 - ab + b^2)$.
Question
Formula: $a^3 - b^3$
Answer
Definition: $(a - b)(a^2 + ab + b^2)$.
Question
Factorise the expression $16a^4 + 54a$.
Answer
$2a(2a + 3)(4a^2 - 6a + 9)$.
Question
Factorise the expression $125a^3 + \frac{1}{8}$.
Answer
$(5a + \frac{1}{2})(25a^2 - \frac{5}{2}a + \frac{1}{4})$.
Question
Factorise the expression $a^3 + b^3 + a + b$.
Answer
$(a + b)(a^2 - ab + b^2 + 1)$.
Question
Factorise the expression $8a^3 - 27b^3$.
Answer
$(2a - 3b)(4a^2 + 6ab + 9b^2)$.
Question
Factorise the expression $2a^7 - 128a$.
Answer
$2a(a + 2)(a - 2)(a^2 - 2a + 4)(a^2 + 2a + 4)$.
Question
How can the identity $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$ be used to show $15^3 - 8^3$ is divisible by 7?
Answer
Since $a - b = 15 - 8 = 7$, the expression $15^3 - 8^3$ has 7 as a factor.
Question
Factorise $x^2 + \frac{a^2 - 1}{a}x - 1$.
Answer
$(x - \frac{1}{a})(x + a)$.
Question
Factorise $a^4 - 7a^2 + 1$ by completing the square.
Answer
$(a^2 + 1 + 3a)(a^2 + 1 - 3a)$.
Question
Factorise $a^3 + 3a^2b + 3ab^2 + 2b^3$.
Answer
$(a + 2b)(a^2 + ab + b^2)$.
Question
Factorise $a^6 - 26a^3 - 27$ using substitution.
Answer
$(a - 3)(a^2 + 3a + 9)(a + 1)(a^2 - a + 1)$.
Question
Factorise the quadratic expression $12x^2 - 35x + 25$.
Answer
$(3x - 5)(4x - 5)$.
Question
Evaluate: $\frac{5.67 \times 5.67 \times 5.67 + 4.33 \times 4.33 \times 4.33}{5.67 \times 5.67 - 5.67 \times 4.33 + 4.33 \times 4.33}$.
Answer
$10$.
Question
What is the simplest form of $\frac{(a + b)(a^2 - ab + b^2)}{a^2b + ab^2}$?
Answer
$\frac{a}{b} - 1 + \frac{b}{a}$.
Question
Find the L.C.M. of $x^2 + 3x + 2$ and $x^2 - 2x - 3$ by factorisation.
Answer
$(x + 1)(x + 2)(x - 3)$.
Question
Find the H.C.F. of $x^2 + 3x + 2$ and $x^2 - 2x - 3$.
Answer
$(x + 1)$.
Question
Factorise $x^2 + \frac{1}{x^2} - 3$ in the form of factors.
Answer
$(x - \frac{1}{x} + 1)(x - \frac{1}{x} - 1)$.
Question
Factorise $a^2 - (2a + 3b)^2$.
Answer
$(3a + 3b)(-a - 3b)$.
Question
Factorise $x(2x - 5) + 3$ into factor form.
Answer
$(x - 1)(2x - 3)$.
Question
What are the factors of $x^4 + x^2 + 1$ (HOTS)?
Answer
$(x^2 + x + 1)(x^2 - x + 1)$.
Question
Factorise the expression $5a^2 - b^2 - 4ab + 7a - 7b$.
Answer
$(a - b)(5a + b + 7)$.