Algebra Class: Factorization and Fractions

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

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

Rearranging gives y = 5 - 2x.

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

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

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

<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

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

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

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

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

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

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

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

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

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

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

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

<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

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

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

<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

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

<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

<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

<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

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

<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

<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

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

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

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

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

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

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

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

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

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

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.

A polynomial with only one term.

A polynomial with two terms.

A polynomial with three terms.

The number that multiplies a variable in a term.

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

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

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

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

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

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

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

The highest power of the variable in a polynomial.

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

The numerical factor in a term.

The term that does not have a variable.

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.

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

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

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