Substitution & biquadratics

Japanese school year: Math I

What you learn

Learn how to factor complex higher-degree polynomials and biquadratic expressions by substituting repeated parts with a single variable or forming a difference of squares. This technique is valuable for simplifying intricate algebraic expressions. Prior mastery of standard factoring formulas and quadratic equations is essential.

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Key points

Use this when the same grouped expression appears repeatedly. Replacing that part with a single variable like $X$ simplifies the expression so standard formulas can be applied.

Use this to factor biquadratic expressions with only even powers, like $x^4$ and $x^2$ . Setting $X = x^2$ reduces the problem to factoring a standard quadratic expression.

ax^4 + bx^2 + c = aX^2 + bX + c \quad (X = x^2)

Use this when simple substitution fails on a fourth-degree expression. Complete the square to form a difference of squares, $(x^2 + k)^2 - (mx)^2$ , then apply the factoring formula.

x^4 + ax^2 + b = (x^2 + k)^2 - (mx)^2

Take care after substituting the original expression back in place of $X$ . Always verify whether each factor inside parentheses can be factored further.

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Practice 1Coming soon