\[ 4p \times (q + r) = 4p \times q + 4p \times r = 4pq + 4pr \]
\[ ab \times (a - b) = ab \times a - ab \times b = a^2b - ab^2 \]
\[ (a + b) \times 7a^2b^2 = a \times 7a^2b^2 + b \times 7a^2b^2 \] \[ = 7a^3b^2 + 7a^2b^3 \]
\[ (a^2 - 9) \times 4a = a^2 \times 4a - 9 \times 4a = 4a^3 - 36a \]
\[ (pq + qr + rp) \times 0 = 0 \]
| First expression | Second expression | Product |
|---|---|---|
| \(a\) | \(b + c + d\) | \( = ab + ac + ad\) |
| \(x + y - 5\) | \(5xy\) | \( = 5x^2y + 5xy^2 - 25xy\) |
| \(p\) | \(6p^2 - 7p + 5\) | \(= 6p^3 - 7p^2 + 5p\) |
| \(4p^2q^2\) | \(p^2 - q^2\) | \( = 4p^4q^2 - 4p^2q^4\) |
| \(a + b + c\) | \(abc\) | \( = a^2bc + ab^2c + abc^2\) |
\[ a^2 \times 2a^{22} \times 4a^{26} \] \[ = 2 \times 4 \times a^{2+22+26} \] \[ = 8a^{50} \]
\[ \frac{2}{3}xy \times \left(-\frac{9}{10}x^2y^2\right) \] \[ = \frac{2}{3} \times \left(-\frac{9}{10}\right) \times x^{1+2} \times y^{1+2} \] \[ = -\frac{18}{30}x^3y^3 = -\frac{3}{5}x^3y^3 \]
\[ -\frac{10}{3}pq^3 \times \frac{6}{5}p^3q \] \[ = -\frac{10}{3} \times \frac{6}{5} \times p^{1+3} \times q^{3+1} \] \[ = -\frac{60}{15}p^4q^4 = -4p^4q^4 \]
\[ x \times x^2 \times x^3 \times x^4 = x^{1+2+3+4} = x^{10} \]
First, simplify the expression:
\[ 3x(4x - 5) + 3 = 12x^2 - 15x + 3 \]
(i) When \(x = 3\):
\[ 12(3)^2 - 15(3) + 3 = 12 \times 9 - 45 + 3 \] \[ = 108 - 45 + 3 = 66 \]
(ii) When \(x = \frac{1}{2}\):
\[ 12\left(\frac{1}{2}\right)^2 - 15\left(\frac{1}{2}\right) + 3 \] \[ = 12 \times \frac{1}{4} - \frac{15}{2} + 3 \] \[ = 3 - 7.5 + 3 = -1.5 \]
First, simplify the expression:
\[ a(a^2 + a + 1) + 5 = a^3 + a^2 + a + 5 \]
(i) When \(a = 0\):
\[ (0)^3 + (0)^2 + (0) + 5 = 5 \]
(ii) When \(a = 1\):
\[ (1)^3 + (1)^2 + (1) + 5 = 1 + 1 + 1 + 5 = 8 \]
(iii) When \(a = -1\):
\[ (-1)^3 + (-1)^2 + (-1) + 5 \] \[ = -1 + 1 - 1 + 5 = 4 \]
\[ p(p - q) + q(q - r) + r(r - p) \] \[ = p^2 - pq + q^2 - qr + r^2 - rp \]
\[ = p^2 + q^2 + r^2 - pq - qr - rp \]
\[ 2x(z - x - y) + 2y(z - y - x) \] \[ = 2xz - 2x^2 - 2xy + 2yz - 2y^2 - 2xy \]
\[ = -2x^2 - 2y^2 + 2xz + 2yz - 4xy \]
First, expand both expressions:
\[ 4l(10n - 3m + 2l) = 40ln - 12lm + 8l^2 \]
\[ 3l(l - 4m + 5n) = 3l^2 - 12lm + 15ln \]
Now subtract:
\[ (40ln - 12lm + 8l^2) - (3l^2 - 12lm + 15ln) \]
\[ = 40ln - 12lm + 8l^2 - 3l^2 + 12lm - 15ln \]Group like terms:
\[ (8l^2 - 3l^2) + (-12lm + 12lm) + (40ln - 15ln) \]
\[ = 5l^2 + 0 + 25ln = 5l^2 + 25ln \]First, expand both expressions:
\[ 4c(-a + b + c) = -4ac + 4bc + 4c^2 \]
\[ 3a(a + b + c) - 2b(a - b + c) \] \[ = 3a^2 + 3ab + 3ac - 2ab + 2b^2 - 2bc \]
\[= 3a^2 + ab + 3ac + 2b^2 - 2bc \]Now subtract:
\[ (-4ac + 4bc + 4c^2) - (3a^2 + ab + 3ac + 2b^2 - 2bc) \]
\[ = -4ac + 4bc + 4c^2 - 3a^2 - ab - 3ac - 2b^2 + 2bc \]
Group like terms:
\[ = -3a^2 - 2b^2 + 4c^2 - ab + (-4ac - 3ac) + (4bc + 2bc) \]
\[ = -3a^2 - 2b^2 + 4c^2 - ab - 7ac + 6bc \]
Developed & Designed by Zahid Qayoom