Learn – Algebra foundations – Multiplication and Factoring

Multiplication without distributive property

Exercises

Problem set

Simplify the following.

$1\times x$
$2\times (-x)$
$-x\times (-y)$
$-3\times (-x)$

Problem set

Simplify the following.

$3\times 6x$
$-2x\times 4$
$4x\times (-5)$
$7x\times (-2)$
$-3x\times (-5)$

Problem set

Simplify the following.

$x\times x$
$1\times x\times x$
$1\times x\times (-x)$
$2\times x\times x \times y$
$1\times x\times y \times y\times x$

Problem set

Simplify the following.

$(-a)\times (-a)$
$2\times (-x)\times (-x)$
$1\times x\times x \times x$
$(-a)\times(-b)\times(-c)$
$4\times (-x)\times (-x) \times (-x)$
$-(-a)$
$3x\times (-y)\times (-x) \times (-4)$

Problem set

Simplify the following.

$2\times x\times y\times (-x)\times(-y)$
$-3\times (-a)\times b\times (-b)\times(a)$
$-5\times p\times (-q)\times p\times (-p)\times (-q)$
$-x\times (-2y)\times 3x\times y$
$-10a\times (-a)\times 2b\times (-2a)\times (-b)\times (-2b)$
$-2\times (-x)\times (-2y)\times x\times (-2y)\times (-2x)$
$-3\times (-p)\times (-2q)\times 2r\times (-2r)\times 2p$

Problem set

Simplify the following.

$2\times 3x^2$
$-3x^2\times 4$
$2xy\times 4xy$
$-3ab^2 \times (-3a^2b)$
$-5pq \times 2pq^2$

Factoring M expression forms

Exercises

Problem set

For each of the following, write as a product of factors and circle the factors.

$xyz$
$2x$
$-3x$
$6xy$
$9x^2$
$10x^2y$

Problem set

For each of the following, write as a product of factors and circle the factors.

$x^3$
$a^4$
$x^3y^2$
$p^2q^3$
$x^2y^2z^3$
$ab^2c^3$
$pq^2p^2$
$xy^2zw^2$

Problem set

For each of the following, write as a product of factors and circle the factors.

$x+2$
$a+b$
$x^2+3$
$pq+6$
$xy+zw$

Problem set

For each of the following, write as a product of factors and circle the factors.

$x(x-1)$
$(p+3)x$
$(a+1)(b+1)$
$(x+4)^2$
$(p-7)^3$
$(a+2)^3(b+3)$

Problem set

For each of the following, write as a product of factors and circle the factors.

$x(x^2+11)$
$x^2(x+5)$
$(x+1)(x+2)(x+3)$
$(2x+1)(3x+2)(4x+3)$
$(x-1)(x+1)(2x+1)(2x-1)$
$(x-2)(x+2)(x^2+2)$
$(2y-1)^2(y+1)$
$(x+1)^2(x^2+1)$
$(x+1)^2(x-1)^2$

Problem set

For each of the following, write as a product of factors and circle the factors.

$yz+4$
$x^2+yz$
$(x+1)(x+2)+6$
$(x+3)^2+6$
$(x-5)^3+10$
$(2x+1)(3x+2)(4x+3)+5$
$(2x+1)(3x+2)(4x+3)+(5x+4)(6x+5)(7x+6)$
$x^3+2x+12$
$x^4+3x^2-4x-12$

Multiplication with distributive property

Exercises

Problem set

For each of the following expressions, multiply the parentheses out.

$x(2x-1)$
$x^2(2x+x^2)$
$p(p-q)$

Problem set

For each of the following expressions, multiply the parentheses out.

$(2x+3)(x+5)$
$(3x+5)(4x+2)$
$(-x-1)(2x+5)$
$(2x-1)(-2x+1)$
$(-3x+2)(-5x-3)$

Problem set

For each of the following expressions, multiply the parentheses out.

$(-4x+3)(-3x+5)$
$(-7x-3)(-2x+3)$
$(x^2-3x+1)(2x-3)$
$(2x^2-1)(3x^2-5x-1)$
$(3x^2-2x-1)(3x^2-2x-1)$

Problem set

Get rid of parentheses in the following by multiplying as appropriate.

$(x^2 + 3)(2x)$
$(2x^2 - 1)(2x - 1)$
$((x+1)(x+2))(2x - 1)$
$\left(\frac{x+1}{x+2}\right)(2x - 1)$
$((x+1)(x+2)+x+3)(2x - 1)$
$\left(\frac{x+1}{x+2}+x+3\right)(2x - 1)$

Factoring A/S expression forms

Exercises

Problem set

For each of the following expressions, write as a product of factors.

$6x+8$
$21x-14$
$9x-15$
$9x+3$
$15x-5$
$12x-8$
$6\times 4\times x-8$

Problem set

For each of the following expressions, write as a product of factors.

$3x^2-2x$
$5p-7p^2$
$4a^2-3ab^2$
$8p^2q^2-6pq^3$
$6x^3-x$
$a^4+a^2+a$
$6x^3-3x$
$p^9+2p^7+p$

Problem set

For each of the following expressions, write as a product of factors.

$p^5+2pq+p$
$x^2(x^3+x)$
$a^2(ab+ac)$
$a^{100}-a^{50}$
$a^{150}-a^{100}+a^{50}$

Problem set

For each of the following expressions, write as a product of factors, with one factor being a negative number.

$-15x-10$
$-16x-12$
$-8x-8$
$-3\times 2\times x-4$
$-5\times 7\times x-7$
$-10\times 9\times x-6$

Problem set

For each of the following expressions, write as a product of factors, with one factor being a negative number.

$-15x+9$
$-24x+16$
$-12x+12$
$-9\times 7\times x+14$
$-5\times 8\times x+8$
$-9\times 14\times x+21$

Problem set

For each of the following expressions, write as a product of factors.

$a(a+2)+2a$
$5a(a+2)+2(a+2)$
$(a-1)(a+2)+2(a-1)$

Problem set

For each of the following expressions, write as a product of factors.

$(p+1)(3p+2)+2(3p+2)$
$3(3p+2)-(1-p)(3p+2)$
$(a-1)^2+3(a-1)$
$(2p-1)^2-(2p-1)$
$(3m-1)^2-(m-1)(3m-1)$

Miscellaneous

Exercises

Problem set

Simplify the following.

$(\sqrt{5} + 5)(1 - \sqrt{5})$
$(\sqrt{5} - 5)(1 + \sqrt{5})$
$(\sqrt{2} + \sqrt{3})\sqrt{6}$ . (Express in terms of $\sqrt{2}$ and $\sqrt{3}$ )
$\frac{\sqrt{5} + 25}{\sqrt{5}}$
$\frac{\sqrt{3}\sqrt{6} + 2}{\sqrt{2}}$

Problem set

Simplify the following.

Simplify: $a(bc)(de) - e(db)(ca)$
$abc+dba = 2$ and $ab = 2$ . What is $d+c$ ?
$19 + b = c + 21$ . What is $3b - 3c$ ?
$ab = \frac{1}{2}$ , $c = d + 2$ . What is the value of $abc - abd$ ?