Intermediate Algebra Pointers and Study Guide

Operations with Powers and Roots

• Multiplication of Powers with the Same Base: When multiplying powers with identical bases, the exponents are added. The formula is:     $a^m \cdot a^n = a^{m+n}$

• Division of Powers with the Same Base: When dividing powers with the same base, the exponents are subtracted. The formula is:     $\frac{a^m}{a^n} = a^{m-n}$

• Power of a Product: A power applied to a product of two or more factors is equal to the product of the powers of each individual factor:     $(abc...)^n = a^n \cdot b^n \cdot c^n ...$

• Power of a Quotient (Fraction): A power of a fraction is equal to the quotient of the powers of the numerator (dividend) and the denominator (divisor):     $(\frac{a}{b})^n = \frac{a^n}{b^n}$

• Raising a Power to a Power: When raising a power to another power, the exponents are multiplied:     $(a^m)^n = a^{mn}$

• Readability of Formulas: All mentioned formulas are reversible; they can be read and executed from left to right or right to left.

• Numerical Example of Combined Power Operations:     $(\frac{2 \times 3 \times 5}{15})^2 = \frac{2^2 \cdot 3^2 \cdot 5^2}{15^2} = \frac{4 \cdot 9 \cdot 25}{225} = \frac{900}{225} = 4$

• Operations with Roots (Arithmetical Roots): These rules assume radicands are positive.

  • Root of a Product: The root of a product of factors is equal to the product of the roots of those factors:         $\sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b}$

  • Root of a Quotient: The root of a quotient is equal to the quotient of the roots of the dividend and divisor:         $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$

  • Raising a Root to a Power: To raise a root to a power, raise the radicand to that power:         $(\sqrt[n]{a})^m = \sqrt[n]{a^m}$

  • Increasing Root Degree: If you increase the degree of a root by $n$ times and simultaneously raise the radicand to the $n$-th power, the value remains unchanged:         $\sqrt[k]{a} = \sqrt[n \times k]{a^n}$

  • Decreasing Root Degree: If you decrease the degree of a root by $n$ times and extract the $n$-th root of the radicand, the value remains unchanged:         $\sqrt[nm]{a^n} = \sqrt[m]{a}$

Widening of Power Notions and Exponents

• Negative Exponents: A power with a negative integer exponent is defined as unity (1) divided by the power of the same number with the absolute value of that exponent:     $a^{-n} = \frac{1}{a^n}$

  • Example: $a^4 : a^7 = a^{4-7} = a^{-3} = \frac{1}{a^3}$

• Zero Exponent: Any non-zero number raised to the zero power is equal to 1:     $a^0 = 1$

  • Examples: $2^0 = 1$, $(-5)^0 = 1$, $(-\frac{3}{5})^0 = 1$

• Fractional Exponents: To raise a real number $a$ to a power with an exponent $\frac{m}{n}$, extract the $n$-th degree root from the $m$-th power of $a$:     $a^{\frac{m}{n}} = \sqrt[n]{a^m}$

  • Example: $64^{\frac{2}{3}} = \sqrt[3]{64^2} = \sqrt[3]{4096} = 16$

Meaningless Expressions and Undefined Values

• Case 1: Division by Zero (Non-zero Divided by Zero):     $\frac{a}{0}$ where $a \neq 0$ does not exist. By definition of division, if $\frac{a}{0} = x$, then $a = 0 \cdot x$, which implies $a = 0$, contradicting the original condition.

• Case 2: Zero Divided by Zero:     $\frac{0}{0}$ represents any number because $0 = 0 \cdot x$ is valid for any value of $x$.

• Case 3: Zero Raised to Zero Power:     $0^0$ is considered to represent any number if standard power rules are extended to zero bases.

• Case Study Example: Solve $\frac{|x|}{x} = 1$

  1. If $x = 0$, the expression is undefined.

  2. If $x > 0$, then $|x| = x$, so $\frac{x}{x} = 1$, which is $1 = 1$ (Valid for all $x > 0$).

  3. If $x < 0$, then $|x| = -x$, so $\frac{-x}{x} = 1$, which is $-1 = 1$ (No solution).

  • Answer: $x > 0$

Formulas of Abridged Multiplication

It is essential to memorize these seven formulas, as they facilitate solving most mathematical problems involving polynomials:

1. Square of a Sum: $(a + b)^2 = a^2 + 2ab + b^2$

2. Square of a Difference: $(a - b)^2 = a^2 - 2ab + b^2$

3. Difference of Squares: $(a + b)(a - b) = a^2 - b^2$

4. Cube of a Sum: $(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$

5. Cube of a Difference: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$

6. Sum of Cubes: $(a + b)(a^2 - ab + b^2) = a^3 + b^3$

7. Difference of Cubes: $(a - b)(a^2 + ab + b^2) = a^3 - b^3$

Factoring Polynomials

• Common Factors: If all terms contain the same expression, it should be factored outside of brackets.

• Grouping Terms: Group terms into brackets to find a shared expression inside, then extract that shared expression as a common factor.

  • Example: $ax + bx + ay + by = (ax + bx) + (ay + by) = x(a + b) + y(a + b) = (x + y)(a + b)$

• Mutually Cancelled Terms: Including new terms that cancel each other out can reveal factoring opportunities.

  • Example: $y^2 - b^2 = y^2 + yb - yb - b^2 = (y^2 + yb) - (yb + b^2) = y(y + b) - b(y + b) = (y + b)(y - b)$

• Factoring Quadratic Trinomials: A quadratic trinomial $ax^2 + bx + c$ can be resolved into first-degree factors by solving the equation $ax^2 + bx + c = 0$. If $x_1$ and $x_2$ are roots, the formula is:     $ax^2 + bx + c = a(x - x_1)(x - x_2)$

  • Example: Resolve $2x^2 - 4x - 6$ to first-degree factors.

  • Solve: $2x^2 - 4x - 6 = 0$ results in roots $x_1 = -1$ and $x_2 = 3$.

  • Result: $2(x + 1)(x - 3)$.

Equations of Higher Degrees

• Reduction to Factors: Some higher-degree equations can be solved by factoring the left-hand side into polynomials no higher than the second degree and setting each to zero.

  • Example: $3x^4 + 6x^3 - 9x^2 = 0$

  • Factor: $3x^2(x^2 + 2x - 3) = 0$

  • Solve: $3x^2 = 0 \rightarrow x_1, x_2 = 0$; $x^2 + 2x - 3 = 0 \rightarrow x_3 = 1, x_4 = -3$.

• Biquadratic Equations: Equations in the shape $ax^{2n} + bx^n + c = 0$ can be reduced using the exchange $x^n = z$.

  • Example: $x^4 - 13x^2 + 36 = 0$

  • Exchange: $x^2 = z$, resulting in $z^2 - 13z + 36 = 0$. Roots are $z_1 = 4, z_2 = 9$.

  • Solve for $x$: $x^2 = 4 \rightarrow x = \pm 2$; $x^2 = 9 \rightarrow x = \pm 3$.

• Cubic Equations: General shape is $ax^3 + bx^2 + cx + d = 0$.

  • Step 1: Find one root ($x_1$) by testing small integers (typically $-2, -1, 0, 1, 2$) that are factors of the constant term $d$.

  • Step 2: Divide the cubic polynomial by the binomial $(x - x_1)$ to obtain a quadratic polynomial.

  • Step 3: Solve the resulting quadratic equation to find the remaining two roots.

  • Example: $x^3 - 3x^2 - 13x + 15 = 0$

  • Trial finds $x = 1$ is a root.

  • Divide $(x^3 - 3x^2 - 13x + 15)$ by $(x - 1)$ to get $(x^2 - 2x - 15)$.

  • Remaining roots of $x^2 - 2x - 15 = 0$ are $x = -3$ and $x = 5$.

Division of Polynomials

• Bezout’s Theorem: If a polynomial $P(x)$ is divided by the linear binomial $(x - b)$, the remainder is a constant $N$ equal to the value of the polynomial evaluated at $x = b$.

  • Example: Dividend $a_0 x^m + a_1 x^{m-1} + ... + a_m$ divided by $x - b$ has a remainder $N = a_0 b^m + a_1 b^{m-1} + ... + a_m$.

• Defining Polynomial Division: To divide political $P$ by polynomial $Q$ means finding a quotient $M$ and a remainder $N$ such that $MQ + N = P$, where the degree of $N$ is less than the degree of $Q$.

• Long Division Example:

  • Dividend: $16a^3 - 4a^2 + 8a + 7$

  • Divisor: $4a^2 - a + 2$

  • 1. Divide $16a^3$ by $4a^2$ to get $4a$.

  • 2. Multiply $4a(4a^2 - a + 2) = 16a^3 - 4a^2 + 8a$.

  • 3. Subtract from dividend: $(16a^3 - 4a^2 + 8a + 7) - (16a^3 - 4a^2 + 8a) = 7$. Wait, transcript states subtracting and moving down term leads to residue $12a^2 - 13a + 7$.

  • 4. Divide $12a^2$ by $4a^2$ to get 3.

  • 5. Multiply $3(4a^2 - a + 2) = 12a^2 - 3a + 6$.

  • 6. Final remainder: $-10a + 1$.

  • Quotient: $4a + 3$. Remainder: $-10a + 1$.

Linear and Quadratic Equations

• Linear Equation in One Unknown: Defined as $ax + b = 0$.

  • If $a \neq 0$, root is $x = -\frac{b}{a}$.

  • If $a = 0, b = 0$, then $x$ can be any number.

  • If $a = 0, b \neq 0$, there is no solution.

• Identical Transformations (Solving Equations):

  1. Replace expressions with identically equal ones (e.g., expanding $(3x+2)^2$).

  2. Transfer terms between sides by changing signs.

  3. Multiply or divide by non-zero constants. (Caution: dividing by an expression that might be zero can lose roots).

  4. Extracting roots or raising to powers (Caution: squaring can create extraneous roots; wrong root extraction can lose roots).

• Quadratic Equations: General form is $ax^2 + bx + c = 0$.

  • Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

  • Reduced Quadratic Equation: If $a = 1$, the form is $x^2 + px + q = 0$. The roots are:         $x = -\frac{p}{2} \pm \sqrt{(\frac{p}{2})^2 - q}$

  • Example: $x^2 + 5x + 6 = 0$. Here $p = 5, q = 6$.

  • $x = -\frac{5}{2} \pm \sqrt{(\frac{5}{2})^2 - 6} = -\frac{5}{2} \pm \sqrt{\frac{1}{4}}$.

  • $x_1 = -2, x_2 = -3$.

Solving Word Problems: Process and Categories

• The 3 R’s and ESP Method:

  • Read: Catch every word carefully.

  • Represent: Choose simple variable representations (e.g., $x$ and $3x$).

  • Relate: Look for key words translating to '=' (is, are, was, make).

  • Equate: Form an equation from the facts.

  • Solve: Manipulate the equation.

  • Prove: Check if roots satisfy the original story.

• I. Number Problems: Relationships are directly stated.

  • Example: One number is two more than thrice another. Sum is 30.

  • Equation: $x + (3x + 2) = 30 \rightarrow 4x = 28 \rightarrow x = 7$. Numbers are 7 and 23.

• II. Consecutive Integers: Whole numbers following without interruption.

  • General: $x, x+1, x+2$.

  • Odd/Even: $x, x+2, x+4$.

  • Example: Sum of three consecutive integers is 90 ($29, 30, 31$). Sum of three consecutive odd integers is 57 ($17, 19, 21$).

• III. Digit Problems: Position matters.

  • 2-digit number: $10x + y$

  • Example: A 2-digit number whose units digit is one more than twice the tens digit ($x$). Sum is 7.

  • Math: $x + (2x + 1) = 7 \rightarrow 3x = 6 \rightarrow x = 2$. Tens digit is 2, Units is 5. Number is 25.

• IV. Age Problems: Ages of all people change at the same rate.

  • Example: Alvin is 20 years older than his son ($x$). In 10 years, he will be twice as old.

  • Future Table: Son: $x + 10$. Alvin: $(x + 20) + 10 = x + 30$.

  • Equation: $x + 30 = 2(x + 10) \rightarrow x = 10$. Present ages: Son 10, Alvin 30.

• V. Work Problems: Rate is defined as $\frac{1}{x}$ where $x$ is time to finish.

  • Formula: $Work = Rate \times Time$.

  • Example: Richard builds a doghouse in 3 days ($Rate = \frac{1}{3}$). Alvin builds it in 6 days ($Rate = \frac{1}{6}$).

  • Equation: $\frac{x}{3} + \frac{x}{6} = 1 \rightarrow 2x + x = 6 \rightarrow x = 2$ days.

• VI. Distance Problems: Uniform motion follows $Distance = Rate \times Time$.

  • Opposite Direction: Mr. Honda ($45\,kph$) and Mr. Toyota ($35\,kph$) leave points $120\,km$ apart.

  • Equation: $45t + 35t = 120 \rightarrow 80t = 120 \rightarrow t = 1.5$ hours.

  • Same Direction: Rabbit ($45\,kph$) leaves 12 mins ($0.2$ hrs) before Panther ($54\,kph$).

  • Equation: $45(x + \frac{1}{5}) = 54x \rightarrow 45x + 9 = 54x \rightarrow 9x = 9 \rightarrow x = 1$ hour.

• VII. Solution Problems: Mixtures involving Solutes and Solvents.

  • Formula: $Amount \times Strength = Solute$.

  • Example: Add $x$ grams of 20% salt to 400g of 10% salt to get 12% solution.

  • Equation: $0.10(400) + 0.20x = 0.12(400 + x) \rightarrow 4000 + 20x = 12(400 + x) \rightarrow 8x = 800 \rightarrow x = 100$ grams.

• VIII. Investment Problems: Involves money at different interest rates (Percent problems).

  • Formula: $p = br$ (Part = Base $\times$ Rate).

  • Example: $P200,000$ total invested. Part $x$ at 4%, the rest at 6%. Total income is $P9,600$.

  • Equation: $0.04x + 0.06(200,000 - x) = 9,600 \rightarrow 4x + 1,200,000 - 6x = 960,000 \rightarrow 2x = 240,000 \rightarrow x = P120,000$.