Miscellaneous Exercise 4: Review of Algebraic Concepts

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Chapter Overview & SLOs

How do we simplify complex factorizations using substitution? Use substitution techniques to convert multi-degree expressions into manageable quadratic forms. Example: Let $a^2 - 5 = x$, then $x^2 - 13x + 36$ factorizes to $(x-4)(x-9)$, then substitute back: $(a^2 - 5 - 4)(a^2 - 5 - 9) = (a^2 - 9)(a^2 - 14) = (a-3)(a+3)(a^2 - 14)$.

How do we factorize expressions with higher powers using successive identities? Use identities like $a^2 - b^2 = (a+b)(a-b)$ and $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$ in succession. Example: $a^{12} - b^{12} = (a^6 - b^6)(a^6 + b^6) = (a^3 - b^3)(a^3 + b^3)(a^2 + b^2)(a^4 - a^2b^2 + b^4)$.

How do we factorize expressions like $2m^4 + m^2n^2 - 3n^4$? Treat as quadratic in $m^2$: $2m^4 + m^2n^2 - 3n^4 = (2m^2 + 3n^2)(m^2 - n^2) = (2m^2 + 3n^2)(m - n)(m + n)$.

How do we find the HCF of large polynomials using long division? Apply the long division method for polynomials. Example: Divide $x^3 + 3x^2 - 8x - 24$ by $x^3 + 3x^2 - 3x - 9$ to find common factors. The remainder determines the HCF. If the remainder is $5x + 15 = 5(x+3)$, then divide the divisor by $x+3$ to find HCF $= x+3$.

How do we find the LCM of multiple algebraic expressions? Factorize each expression completely, then take the product of all common and uncommon factors with their highest powers. Example: $x^2+3x+2 = (x+1)(x+2)$, $x^2+4x+3 = (x+1)(x+3)$, LCM $= (x+1)(x+2)(x+3)$.

How do we simplify rational products and quotients? Factorize all numerators and denominators completely, then cancel common factors across terms. Example: $\frac{x^2 - 4}{x^2 - 1} \times \frac{x^2 + 3x + 2}{x^2 - 2x - 8} = \frac{(x-2)(x+2)}{(x-1)(x+1)} \times \frac{(x+1)(x+2)}{(x-4)(x+2)} = \frac{x-2}{(x-1)(x-4)}$.

How do we calculate square roots of complex rational expressions? Implement the division algorithm for square roots. Example: Find square root of $\frac{9a^2}{x^2} - \frac{6a}{5x} + \frac{101}{25} - \frac{4x}{15a} + \frac{4x^2}{9a^2}$.

• Step 1: Rearrange in descending powers: $\frac{9a^2}{x^2} - \frac{6a}{5x} + \frac{101}{25} - \frac{4x}{15a} + \frac{4x^2}{9a^2}$
• Step 2: First term: $\sqrt{\frac{9a^2}{x^2}} = \frac{3a}{x}$
• Step 3: Double the quotient, find next term $= -\frac{1}{5}$
• Step 4: Continue to get $\frac{2x}{3a}$
• Final square root: $\frac{3a}{x} - \frac{1}{5} + \frac{2x}{3a}$

How do we solve for unknown constants $a$ and $b$ for perfect squares? Perform division method including the unknowns, then set the remainder to zero. Example: For $x^4 + ax^3 + bx^2 - 8x + 4$ to be a perfect square, assume square root $= x^2 + px + q$, square it, compare coefficients to find $a$ and $b$.

Remember the substitution method: In Q2, Q4, and Q6, you must remember to substitute the original variable back into the factored expression to get the final answer.

These solutions are strictly aligned with the Student Learning Outcomes (SLOs) for the FBISE 2026 annual examination.

• How do we factorize expressions with higher powers using successive identities? Use substitution techniques (e.g., let $a^2 - 5 = x$ to simplify $x^2 - 13x + 36$) and identities like $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$ and $a^2 - b^2 = (a+b)(a-b)$ in succession to factorize expressions like $a^{12} - b^{12}$ completely.
• How do we find the LCM of multiple algebraic expressions? Factorize each expression completely (e.g., $x^2+3x+2 = (x+1)(x+2)$, $x^2+4x+3 = (x+1)(x+3)$) and take the product of all common and uncommon factors with their highest powers to find the LCM.
• How do we simplify rational products and quotients? Factorize all numerators and denominators completely, then cancel common factors across terms to reduce the expression to its simplest form, being careful with sign cancellations and domain restrictions.
• How do we calculate square roots of complex rational expressions? Implement the division algorithm for square roots, determining terms of the quotient (e.g., $\frac{3a}{x} - \frac{1}{5} + \frac{2x}{3a}$) through step-by-step subtraction of squared terms, and solve for unknown constants $a$ and $b$ by ensuring the remainder of a division for a perfect square equals zero.

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