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Let’s factorise, say and write the answers as quickly as possible.
a) $ax + bx =$ ………………..
b) $ax – ay =$ ………………..
c) $y^2 – y =$ ………………..
d) $2x^3 + x^2 =$ ………………..
e) $x^2 – x^3 =$ ………………..
f) $x^2y + xy^2 =$ ………………..
a) If $(x + y) = 5$ and $xy = 3$, find the value of $x^2 + y^2$.
b) If $(a – b) = 4$ and $ab = 2$, find the value of $a^2 + b^2$.
c) If $left(x + frac{1}{x}right) = 3$, find the value of $x^2 + frac{1}{x^2}$.
d) If $left(p – frac{1}{p}right) = 7$, find the value of $p^2 + frac{1}{p^2}$.
Remember:
$(x + y)^2 = x^2 + 2xy + y^2$
and, $x^2 + y^2 = (x + y)^2 – 2xy$
Let’s find the products using the formula $(a + b)(a – b) = a^2 – b^2$.
Hint: $99 times 101 = (100 – 1) times (100 + 1) = 100^2 – 1^2 = 10000 – 1 = 9999$
a) $19 times 21$
b) $49 times 51$
c) $78 times 82$
d) $102 times 98$
Let’s apply the appropriate formula to find the products.
a) $(a + 3)(a – 3)$
b) $(2x + 1)(2x – 1)$
c) $(4 + 3p)(4 – 3p)$
d) $(2x – 3y)(2x + 3y)$
e) $(x^2 – y^2)(x^2 + y^2)$
f) $(2x^2 + 5y^2)(2x^2 – 5y^2)$
g) $(a + b + c)(a + b – c)$
h) $(x – y + z)(x + y + z)$
i) $(p – q – r)(p + q – r)$
j) $(x + y)(x – y)(x^2 + y^2)$
k) $(x + 2)(x – 2)(x^2 + 4)$
l) $(2a + y)(2a – y)(4a^2 + y^2)$
hkjhj
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a) Find the squares of
(i) $2a + 1$
(ii) $x – 3y$
(iii) $a – frac{1}{a}$
(iv) $x + frac{1}{x}$
b) Expand
(i) $(3x – 1)^2$
(ii) $left(2y + frac{1}{2y}right)^2$
Let’s say and write the expanded forms of these expressions.
| Expressions | Expanded forms |
| a) $(x + y)^2$ | |
| b) $(a + 1)^2$ | |
| c) $(p + 2)^2$ | |
| d) $(y + 3)^2$ | |
| e) $(x – y)^2$ | |
| f) $(m – 1)^2$ | |
| g) $(a – 2)^2$ | |
| h) $(p – 3)^2$ |
Let’s say and write the following products in the forms of difference of two squared terms.
a) $(x + y)(x – y) =$ ………………..
b) $(x + 1)(x – 1) =$ ………………..
c) $(p + 2)(p – 2) =$ ………………..
d) $(m + 3)(m – 3) =$ ………………..
e) $left(a + frac{1}{a}right)left(a – frac{1}{a}right) =$ ………………..
f) $left(x + frac{2}{x}right)left(x – frac{2}{x}right) =$ ………………..
g) $left(frac{3}{p} + pright)left(frac{3}{p} – pright) =$ ………………..
h) $left(frac{4}{m} + mright)left(frac{4}{m} – mright) =$ ………………..
Write the degrees of the following polynomials.
(i) $3x^2$
(ii) $2a^2bc$
(iii) $2x^3 – 4x^2 + 7x$
(iv) $x^3y^2 + x^2y^2 – xy$
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Find the squares of: a) $3x + 2y$ b) $left(x – frac{1}{x}right)$
Which of the following expressions are polynomials? Write with reason.
(i) $x^2 + frac{1}{2}$
(ii) $y^2 + frac{1}{y^2}$
(iii) $sqrt{3}x – 2$
(iv) $2sqrt{x} + 5$
