Exploring Algebraic Identities

Using the identity $(a + b)^2 = a^2 + 2ab + b^2$, expand the following:

(i) $(7x + 4y)^2$

(ii) $\left(\frac{7}{5}x + \frac{3}{2}y\right)^2$

(iii) $(2.5p + 1.5q)^2$

(iv) $\left(\frac{3}{4}s + 8t\right)^2$

(v) $\left(x + \frac{1}{2y}\right)^2$

(vi) $\left(\frac{1}{x} + \frac{1}{y}\right)^2$

Using the same identity, find the values of the following:

(i) $(64)^2$

(ii) $(105)^2$

(iii) $(205)^2$

Factor completely:

(i) $9x^2 + 24xy + 16y^2$

(ii) $4s^2 + 20st + 25t^2$

(iii) $49x^2 + 28xy + 4y^2$

(iv) $64p^2 + \frac{32}{3}pq + \frac{4}{9}q^2$

*(v) $3a^2 + 4ab + \frac{4}{3}b^2$

*(vi) $\frac{9}{5}s^2 + 6sv + 5v^2$

(Hint: 2 was taken out as a common factor in Example 7. Is it possible to do something similar in Exercises (v) and (vi) above?)

Find the values of the following using the identity $(a - b)^2 = a^2 - 2ab + b^2$.

(i) $(79)^2$

(ii) $(193)^2$

(iii) $(299)^2$

Find the following squares using one of the above identities. Determine which of these identities will make these calculations easier.

(i) $117^2$

(ii) $78^2$

(iii) $198^2$

(iv) $214^2$

(v) $1104^2$

(vi) $1120^2$

Factor using suitable identities:

(i) $16y^2 - 24y + 9$

(ii) $\frac{9}{4}s^2 + 6st + 4t^2$

(iii) $\frac{m^2}{9} + \frac{mk}{3} + \frac{k^2}{4} + 3nk + 2mn + 9n^2$

(iv) $\frac{p^2}{16} - 2 + \frac{16}{p^2}$

(v) $9a^2 + 4b^2 + c^2 - 12ab + 6ac - 4bc$

Expand the following using the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$:

(i) $(p + 3q + 7r)^2$

(ii) $(3x - 2y + 4z)^2$

Is this an identity?

$(a + b - c)^2 + (a - b + c)^2 + (a - b - c)^2 = 2a^2 + 2b^2 + 2c^2$.

Fill in the blanks to complete the following identities:

Select and use the identity that will help you find the following products without multiplying directly:

(i) $(41)^2$

(ii) $(27)^2$

(iii) $(23 \times 17)$

(iv) $(135)^2$

(v) $(97)^2$

(vi) $(18 \times 29)$

(vii) $(34 \times 43)$

(viii) $(205)^2$

Factor the following:

(i) $9a^2 + b^2 + 4c^2 - 6ab + 12ac - 4bc$

(ii) $16s^2 + 25t^2 - 40st$

(iii) $r^2 - r - 42$

(iv) $49g^2 + 14gh + h^2$

(v) $64u^2 + 121v^2 + 4w^2 - 176uv - 32uw + 44vw$

Simplify the following rational expressions assuming that the expressions in the denominators are not equal to zero:

(i) $\frac{3p^2 - 3pq - 18q^2}{p^2 + 3pq - 10q^2}$

(ii) $\frac{n^3 - 3n^2m + 3nm^2 - m^3}{5m^2 - 10mn + 5n^2}$

(iii) $\frac{w^3 - v^3 + x^3 + 3wvx}{w^2 + v^2 + x^2 - 2wv - 2vx + 2wx}$

(iv) $\frac{4y^2 - 20yz + 25z^2}{(25z^2 - 4y^2)}$

(v) $\frac{(x^2 + x - 6)(x^2 - 7x + 12)}{(x^2 - 6x + 8)(x^2 - 9)}$

(vi) $\frac{p^4 - 16}{p^2 - 4p + 4}$

Use suitable identities to find the following products:

(i) $(-3x + 4)^2$

(ii) $(2s + 7)(2s - 7)$

(iii) $\left(p^2 + \frac{1}{2}\right)\left(p^2 - \frac{1}{2}\right)$

(iv) $(2n + 7)(2n - 7)$

(v) $(s - 2t)(s^2 + 2st + 4t^2)$

(vi) $\left(\frac{1}{2r} - 4r\right)^2$

(vii) $(-3m + 4k - l)^2$

(viii) $\left(x - \frac{1}{3}y\right)^3$

(ix) $\left(\frac{7}{2}k - \frac{2}{3}m\right)^3$

Find the values using suitable identities:

(i) $17 \times 21$

(ii) $104 \times 96$

(iii) $24 \times 16$

(iv) $147^3$

(v) $199^3$

(vi) $127^3$

(vii) $(-107)^3$

(viii) $(-299)^3$

Factor the following algebraic expressions:

(i) $4 y^2 + 1 + \frac{1}{16 y^2}$

(ii) $9m^2 - \frac{1}{25n^2}$

(iii) $27b^3 - \frac{1}{64b^3}$

(iv) $x^2 + \frac{5x}{6} + \frac{1}{6}$

(v) $27u^3 - \frac{1}{125} - \frac{27u^2}{5} + \frac{9u}{25}$

(vi) $64y^3 + \frac{1}{125}z^3$

(vii) $p^3 + 27q^3 + r^3 - 9pqr$

(viii) $9m^2 - 12m + 4$

(ix) $9x^3 - \frac{8}{3}y^3 + \frac{z^3}{3} + 6xyz$

(x) $4x^2 + 9y^2 + 36z^2 + 12xz + 36yz + 24xy$

(xi) $27u^3 - \frac{1}{216} - \frac{9u^2}{2} + \frac{u}{4}$

Simplify the following:

(i) $\frac{4x^2 + 4x + 1}{4x^2 - 1}$

(ii) $\frac{9(3a^3 - 24b^3)}{9a^2 - 36b^2}$

(iii) $\frac{s^3 + 125t^3}{s^2 - 2st - 35t^2}$

Note: Assume that the denominators are not equal to 0.

Find possible expressions for the length and breadth of each of the following rectangles whose areas are given by the following expressions in square units.

(i) $25a^2 - 30ab + 9b^2$

(ii) $36s^2 - 49t^2$

Find possible expressions for the length, breadth, and heights of each of the following cuboids whose volumes are given by the following expressions in cubic units.

(i) $6a^2 - 24b^2$

(ii) $3ps^2 - 15ps + 12p$

The village playground is shaped as a square of side 40 metres. A path of width $s$ metres is created around the playground for people to walk. Find an expression for the area of the path in terms of $s$.

If a number plus its reciprocal equals $\frac{10}{3}$, find the number.

A rectangular pool has area $2x^2 + 7x + 3$ square hastas. If its width is $2x + 1$ hastas, find its length. Hasta was a unit used to measure length.

If both $x - 2$ and $x - \frac{1}{2}$ are factors of $px^2 + 5x + r$, show that $p = r$.

If $a + b + c = 5$ and $ab + bc + ca = 10$, then prove that $a^3 + b^3 + c^3 - 3abc = -25$.

By factoring the expression, check that $n^3 - n$ is always divisible by 6 for all natural numbers $n$. Give reasons.

Find the value of

(i) $x^3 + y^3 - 12xy + 64$, when $x + y = -4$

(ii) $x^3 - 8y^3 - 36xy - 216$, when $x = 2y + 6$