Factorization, Algebraic Fractions & Change of Subject PDF
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This document provides practice questions and examples on factorization, algebraic fractions, and changing the subject of a formula. It is suitable for secondary school students.
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## Factorization, Algebraic Fractions and Change of Subject ### 1.1 Factorization by Taking Out Common Factors #### What Did I Learn? * **Factorization:** * The reverse process of expansion of expressions. * For example: $x^2 - 7x + 12 = (x-3)(x-4)$. * **Factorizing expressions by taking...
## Factorization, Algebraic Fractions and Change of Subject ### 1.1 Factorization by Taking Out Common Factors #### What Did I Learn? * **Factorization:** * The reverse process of expansion of expressions. * For example: $x^2 - 7x + 12 = (x-3)(x-4)$. * **Factorizing expressions by taking out common factors:** * For example: * $2xy+3x = x(2y+3)$. * $(b-d)c(b-d)a = (b-d)(ca)$. #### What Can I Do? * **Distinguish between factorization or expansion.** * Determine whether each of the following processes is factorization or expansion: * `hx-hy+kx-ky → (h+k)(x-y)` * `(a-2)(a+1)(a-1) → a^2 - 2a^2 - a + 2` * **Factorize expressions with 2 terms or 3 terms.** * Factorize the following expressions: * `ax + 9ay` * `-10b^4 - 2b^2` * `4hk - 8hk^2` * `-25uv^3 + 20u^2v^2` * **Factorize the following expressions.** * `3n² + 6mn - 15n` * `8r^3s - 12r^3s^2 + 20r^4` ### 1.2 Factorization by Grouping Terms #### What Did I Learn? * **Factorizing expressions by grouping terms:** * Group the terms with common factors in pairs. * Take out the common factor in each group. * Take out the common factor among the groups. #### What Can I Do? * **Factorize expressions with 4 terms.** * Factorize `5ab + 2a`. * Hence, factorize `5ab + 2a - 10b - 4`. * **Factorize `4xy-y`.** * **Hence, factorize `4xy-y-1+4x`.** * **Factorize the following expressions:** * `h² - hk - 5h + 5k` * `10v - 20uv + 60u^3 - 30u^2` * **Factorize `-2ax² - 2ayz - 2bx² - 2byz`.** ### 10.2 Difference of Two Squares #### What Did I Learn? * The identity of the difference of two squares: * $(a+b)(a-b) = a^2-b^2$ #### What Can I Do? * **Expand the following expressions:** * `(c+10)(c-10)` * `(4 + 3n)(4-3n)` * **Expand the following expressions:** * `(9y+2x)(2x-9y)` * `(1/2 + 2r)^2` * `(-2r/5)^2` * **Expand the following expressions:** * `(-2a+7)(-2a-7)` * `(5/u - v)^2` * **Expand the following expressions:** * `-3(2b + 1)(2b-1)` * `(3x-6y)(6y+3x)` ### 10.3 Perfect Square #### What Did I Learn? * The identities of the perfect square: * $(a+b)^2 = a^2 + 2ab + b^2$ * $(a-b)^2 = a^2 -2ab + b^2$ #### What Can I Do? * **Expand the following expressions:** * `(7-k)^2` * `(4a+3b)^2` * **Expand the following expressions:** * `(xy-2)^2` * `\left(6+h^2\right)^2` * **Expand the following expressions:** * `\left(\cfrac{c}{2} + 3\right)^2` * `\left(\cfrac{2r}{5} + 5\right)^2` * **Expand the following expressions:** * `(-5p+2q)^2` * `\left(\cfrac{3}{u} - v\right)^2 ` * **Expand the following expressions:** * `3(2x-y)^2` * `-4(-3m+7n)^2` ### 10.1 Meaning of Identities #### What Did I Learn? * **Identity:** An equation that can be satisfied by all values of its unknown(s). * **Terms in an identity:** They are the same on both sides. #### What Can I Do? * **Prove that an equation is an identity.** * Prove that each of the following equations is an identity: * `x(x + 7) - 5 = x^2 + 7x - 5` * `-2(3x-1) = 3(1-2x)-1` * `(4x-3)-(x-1) = (3x - 5)` * `(8x- 5) = 1/3 (8x-5)` * **Prove that an equation is not an identity.** * Prove that each of the following equations is not an identity: * `(x-2) + 2 = 2x + 1` * `3x^2 - 5(2-x) = 10 + x(3x+5)` ### 5.1 Simplification of Expressions in Index Notation #### What Did I Learn? * **Laws of indices:** * `a^m \times a^n = a^(m+n)` * `a^m / a^n = a^(m-n)` * `(a^m)^n = a^(m \times n)` * `(ab)^n = a^n \times b^n` * `(a/b)^n = a^n / b^n` #### What Can I Do? * **Find the value of expressions.** * Without using a calculator, find the value of each of the following expressions: * `3^4` * `(-1/2)^3` * **Use index notation to represent expressions.** * Use index notation to represent each of the following expressions: * `3 \times x \times x \times x \times x \times x \times 3` * `a \times x \times b \times x \times a \times x \times a \times x \times b \times a` * **Simplify each of the following expressions and represent the answer in index notation.** * `5^7 \times 5^2` (base: 5) * `-11^3 \times 11^8` (base: 11) * `(3^5)^3` (base: 3) * `(2^4)^2 \times 2^5` (base: 2) ### 5.2 Concept of Polynomials #### What Did I Learn? * **Number of terms (monomials, binomials, trinomials and polynomials):** * For example: The number of terms of `4x² + x - 5` is 3. * **Coefficient of a term, constant term:** * For Example: The coefficient of the `x` term and the constant term of `4x^² + x - 5` are 1 and -5 respectively. * **Degree of a polynomial, arrangement of terms in a polynomial:** * For example: The degree of `4x^2 + x - 5` is 2. It is arranged in descending powers of `x`. #### What Can I Do? * **Distinguish monomials and polynomials from algebraic expressions.** * Determine whether each of the following expressions is a monomial or not: * $-2x$ * $\frac{5h}{k}$ >* **Monomial**: an expression that is the product of a number and one or more variable terms. >* **Polynomial**: an expression that is the sum of one or more monomials. * **Determine whether each of the following expressions is a polynomial or not.** * `-1 - 2m + \frac{3}{m}` * `3a^2 + 5a - 1` * **Identify numbers of terms, degrees, coefficients and constant terms of polynomials.** * Consider the polynomial `5 - 2b^4 + b^3`. * Write down the number of terms of the polynomial. * Write down the degree of the polynomial. * Write down the coefficient of the term `b^4` of the polynomial. * Write down the constant term of the polynomial. * **Consider the polynomial `2xy^n - y^2 - 7x + 3` , where `n` is a positive integer.** * Write down the coefficient of the `y^2` term of the polynomial. * If the degree of the polynomial is 5, write down the value of `n`. ### 5.3 Addition and Subtraction #### What Did I Learn? * **Like terms and unlike terms:** * For example: `-3ab^2c` and `4abc` are like terms. `6pq^2` and `8pq^3` are unlike terms. * **Like terms in polynomials can be added or subtracted.** * Same indices for respective variables * For example: `mn + 5 - 3mn = mn - 3mn + 5 = -2mn + 5` #### What Can I Do? * **Add and subtract like terms in polynomials.** * Simplify each of the following polynomials: * `-6u^2 - 3u + 2u^2 - u` * `ab^2 + 4b - 6ab^2` * `5a - 3b - 8b - a` * `3hk^7h^2 - 8hk + 2h^2 + 4k^2` * **Add and subtract polynomials.** * Complete each of the following operations: * `6x^2 - x + 3` * `5u^3 - 3uv` * `+ x^2 - 5` * `-u^3 + 2uv - 4uv^2` * **Simplify each of the following expressions:** * `(4a - 7b) + (2a^2 - 3b^2 - 3b - 5)` * `(2 - 5m + 3m^2) - (3 - m + 2m^2)` * `(3xy - 4x) - (-2y + 5x - xy)` * `(2r^2 - 7rs - 6s^2) + (-r^2 + 3rs + 6s^2)` ### 1.5 Change of Subject of a Formula #### What Did I Learn? * **Subject of a formula:** It is the single variable on its own on one side of the equal sign, while the other side of the equal sign does not contain the variable. * For example: `V` is the subject of the formula `V=B \times h`. #### What Can I Do? * **Make a letter the subject of a formula.** * In each of the following, make the letter in brackets the subject of the formula: * `F=kx` [k] * `K = \frac{P}{2m}` [m] * `P = 2(a+b)` [b] * `2x + y - 5 = 0` [y] * **Make `x` the subject of the formula `\frac{x}{6} + \frac{y}{3} = 1`** * **Make a letter the subject of a formula with like terms.** * Make `s` the subject of the formula `sx-5 = sy`. * **Make `u` the subject of the formula `2(u + mv) = mu`** ### 5.4 Multiplication #### What Did I Learn? * **Multiplying a monomial by a polynomial:** * For example: `a(b+c+d) = ab+ac+ad` * **Multiplying a binomial by a polynomial:** * For example: `(a+b)(u+v+w)=a(u+v+w)+b(u+v+w) = au+av+aw+bu+bv+bw` #### What Can I Do? * **Multiply monomials by polynomials.** * Expand each of the following expressions: * `4x(3x-1)` * `-m²(5-2m^2)` * `k(4h+3k)` * `(b-4c)(-bc)` * **Expand each of the following expressions:** * `(a^3-5a+2)(2a)` * `-3u(u+2v-6)` * **Multiply binomials by polynomials.** * Expand each of the following expressions: * `(y-3)(4y+3)` * `(-2p+q)(2q-7p)` * `(n-2)(n^2-4n+1)` * `(4r-s)(4r+s-3)`