Algebra Class: Factorization and Fractions

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Questions and Answers

How do you isolate k in the formula F = kx?

To isolate k, divide both sides by x, resulting in k = rac{F}{x}.

What is the process to make m the subject in the equation K = rac{P}{2m}?

Multiply both sides by 2m and divide by K, leading to m = rac{P}{2K}.

If you want to solve for y in the equation 2x + y - 5 = 0, what would be the result?

Rearranging gives y = 5 - 2x.

Expand the expression 4x(3x-1) and write the resulting polynomial.

<p>The expanded expression is <code>12x^2 - 4x</code>.</p> Signup and view all the answers

What do you obtain when you expand the binomial (y-3)(4y+3)?

<p>The result is <code>4y^2 - 12y - 9</code>.</p> Signup and view all the answers

What is the degree of the polynomial 4x^2 + x - 5?

<p>2</p> Signup and view all the answers

Is -2x a monomial? Explain your reasoning.

<p>Yes, it is a monomial because it is the product of a number and a single variable.</p> Signup and view all the answers

Determine whether the expression 3a^2 + 5a - 1 is a polynomial.

<p>Yes, it is a polynomial.</p> Signup and view all the answers

From the polynomial 5 - 2b^4 + b^3, how many terms does it have?

<p>3</p> Signup and view all the answers

In 2xy^n - y^2 - 7x + 3, what is the coefficient of the term y^2?

<p>-1</p> Signup and view all the answers

Simplify the expression -6u^2 - 3u + 2u^2 - u.

<p>-4u^2 - 4u</p> Signup and view all the answers

What is the subject of the formula p = 2q + r, isolating q?

<p>q = (p - r)/2</p> Signup and view all the answers

If the degree of 2xy^n - y^2 - 7x + 3 is 5, what is the value of n?

<p>5</p> Signup and view all the answers

Expand the expression (7-k)^2.

<p>The expanded form is $49 - 14k + k^2$.</p> Signup and view all the answers

Prove that the equation x(x + 7) - 5 = x^2 + 7x - 5 is an identity.

<p>Both sides simplify to the same expression, $x^2 + 7x - 5$.</p> Signup and view all the answers

Use index notation to represent the expression 3 imes x imes x imes x imes x imes x imes 3.

<p>The index notation is $9x^5$.</p> Signup and view all the answers

Expand the expression (-5p + 2q)^2.

<p>The expanded form is $25p^2 - 20pq + 4q^2$.</p> Signup and view all the answers

Simplify the expression 5^7 imes 5^2 and represent the answer in index notation.

<p>The simplified form is $5^9$.</p> Signup and view all the answers

Why is the equation (x-2) + 2 = 2x + 1 not an identity?

<p>It does not hold true for all values of $x$; for example, $x=1$ yields different results.</p> Signup and view all the answers

Expand the expression rac{c}{2} + 3)^2.

<p>The expanded form is $ rac{c^2}{4} + 3c + 9$.</p> Signup and view all the answers

What is the coefficient of the term x in the polynomial 4x^2 + x - 5?

<p>The coefficient of the term $x$ is 1.</p> Signup and view all the answers

What is factorization and provide an example?

<p>Factorization is the reverse process of expansion of expressions. For example, $x^2 - 7x + 12$ can be factorized to $(x-3)(x-4)$.</p> Signup and view all the answers

How would you factor the expression $2xy + 3x$?

<p>The expression can be factored as $x(2y + 3)$.</p> Signup and view all the answers

Explain the difference between factorization and expansion with examples.

<p>Factorization reduces expressions into products of factors, as in $hx - hy + kx - ky o (h+k)(x-y)$, while expansion increases products to sums, as in $(a-2)(a+1)(a-1) o a^2 - 2a^2 - a + 2$.</p> Signup and view all the answers

Demonstrate the grouping method by factorizing $h^2 - hk - 5h + 5k$.

<p>By grouping, factor the first two terms as $h(h-k)$ and the last two as $5(k-1)$, leading to $(h+5)(h-k)$ after factoring out common terms.</p> Signup and view all the answers

Identify the identity of the difference of two squares and provide an example.

<p>The identity is $(a+b)(a-b) = a^2 - b^2$. For example, $(c+10)(c-10) = c^2 - 100$.</p> Signup and view all the answers

What is the result of expanding $(4 + 3n)(4 - 3n)$?

<p>The result of expanding this expression is $16 - 9n^2$.</p> Signup and view all the answers

Explain the identity of a perfect square and provide an example.

<p>The identity is $(a+b)^2 = a^2 + 2ab + b^2$. For example, $(x + 2)^2$ expands to $x^2 + 4x + 4$.</p> Signup and view all the answers

How do you factor the expression $-2ax^2 - 2ayz - 2bx^2 - 2byz$?

<p>Factor out $-2$ to get $-2(ax^2 + ayz + bx^2 + byz)$, and further group as $-2((a+b)(x^2 + yz))$.</p> Signup and view all the answers

Flashcards

Factorization

The process of breaking down an expression into its factors, like finding the numbers that multiply to give a specific number.

Factorization by Taking Out Common Factors

Factoring out the greatest common factor shared by all the terms in an expression.

Factorization by Grouping Terms

Grouping terms with common factors to factorize expressions with 4 or more terms.

Difference of Two Squares

A special form of factorization where the expression is the difference of two perfect squares, like (a^2 - b^2).

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Perfect Square

The result of squaring a binomial, like (a+b)^2 or (a-b)^2, which produces a specific pattern.

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Expansion

The reverse of factorization, where factors are multiplied out to expand the expression.

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Square of a Sum

The identity (a+b)^2 = a^2 + 2ab + b^2.

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Square of a Difference

The identity (a-b)^2 = a^2 - 2ab + b^2.

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Identity (in algebra)

An equation that holds true for all possible values of its variables.

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Polynomial

A mathematical expression consisting of one or more terms, each of which is a product of a constant and one or more variables raised to non-negative integer powers.

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Monomial

A polynomial with only one term.

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Binomial

A polynomial with two terms.

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Trinomial

A polynomial with three terms.

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Coefficient

The number that multiplies a variable in a term.

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Constant term

The term in a polynomial that does not have a variable.

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Product of powers rule

A rule that states that when multiplying powers with the same base, you add their exponents.

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What is a monomial?

An expression containing only one term, which is a product of a constant and one or more variables raised to non-negative integer powers.

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Making a Letter the Subject of a Formula

The process of rearranging a formula to isolate a specific variable on one side of the equation.

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What is a polynomial?

An expression that is the sum of one or more monomials.

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Multiplying a Monomial by a Polynomial

When multiplying a monomial by a polynomial, you distribute the monomial to each term inside the polynomial.

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Multiplying a Binomial by a Polynomial

When multiplying a binomial by a polynomial, you distribute each term from the binomial to every term in the polynomial.

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What is the degree of a polynomial?

The highest power of the variable in a polynomial.

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Expanding Expressions

Expanding an expression means removing parentheses by applying the distributive property.

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What is a coefficient?

The numerical factor in a term.

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What is a constant term?

The term that does not have a variable.

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Making a Letter the Subject of a Formula with Like Terms

To make a letter the subject of a formula with like terms, you need to group the terms containing the desired variable on one side of the equation.

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What are like terms?

Terms that have the same variables raised to the same powers.

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How do you simplify polynomials?

The process of combining like terms in a polynomial by adding or subtracting their coefficients.

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What is the subject of a formula?

A variable that is isolated on one side of the equation, while the other side does not contain that variable.

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Study Notes

Factorization

  • Factoring expressions: Finding expressions that multiply together to produce a given expression. This is the reverse of expansion.
  • Common factors: Identify and factor out common factors from expressions.
  • Binomial factorization: Factoring expressions with two terms, paying attention to common factors within the terms.
  • Trinomial factorization: Factoring expressions with three terms using various methods to determine factors.
  • Factorization by Grouping: Factoring expressions with four or more terms by grouping terms with common factors.

Algebraic Fractions

  • Fractions containing variables: Simplify and manipulate algebraic fractions with common factors and terms across the numerator and denominator or by taking out common factors.

Change of Subject

  • Subject of a formula: Isolating a specified variable in a formula. Manipulate equations by applying algebraic operations (addition, subtraction, multiplication, division) to both sides to isolate the desired variable.
  • Rearranging formulas: Change the subject of a formula to isolate a specific variable.

Polynomials

  • Laws of indices: Rules for manipulating exponents (e.g., am * an = am+n, (am)n = amn).

  • Index notation: Expressing numbers and variables involving exponents in a compact form (e.g., 3x³).

  • Simplifying expressions: Using index laws to simplify expressions with exponents and variables.

  • Adding and subtracting polynomials: Combine like terms in expressions to simplify.

  • Multiplying polynomials: Multiplying monomials by polynomials, binomials by polynomials.

  • Polynomial identities: Demonstrate using polynomial identities.

  • Difference of two squares: A special case of factoring expressions where subtracting two squares equals the product of a sum and a difference of two terms.

  • e.g. (x2-y2) = (x+y)(x-y)

  • Concept of Polynomials:

  • Number of terms (monomials, binomials, trinomials and polynomials)

  • Coefficient of a term, constant term

  • Degree of a polynomial, arrangement of terms in a polynomial

  • Simplifying expressions with variables and exponents: Apply rules and properties to condense or rewrite algebraic expressions.

  • Expanding: Expanding expressions using distributive properties, expanding equations with parentheses, using identities to express expansions algebraically, e.g., (a+b)(a-b)=a2−b2

General

  • Evaluation of Polynomials and Expressions: Substitution into expressions to find variable values from known values.
  • Integrated Questions: Combined problems from different topics in algebra.

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Factorization, Algebraic Fractions & Change of Subject PDF

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