1Cademy - Squaring $$(4u^3 + 1)^2$$ Using the Binomial Squares Pattern

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Binomial Squares Pattern

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Squaring $(4u^3 + 1)^2$ Using the Binomial Squares Pattern

Apply the Binomial Squares Pattern to multiply $(4u^3 + 1)^2$ . Use the addition form $(a + b)^2 = a^2 + 2ab + b^2$ , where $a = 4u^3$ and $b = 1$ .

Step 1 — Square the first term: $(4u^3)^2 = 16u^6$ . Square the coefficient: $4^2 = 16$ . Apply the Power Property to the variable: $(u^3)^2 = u^{3 \cdot 2} = u^6$ .

Step 2 — Square the last term: $1^2 = 1$ .

Step 3 — Double their product: $2 \cdot (4u^3) \cdot 1 = 8u^3$ .

Assemble the three pieces into a trinomial:

$(4u^3 + 1)^2 = 16u^6 + 8u^3 + 1$

This example shows the pattern applied to a binomial containing a higher-degree term ( $4u^3$ ). Squaring a term like $4u^3$ requires two exponent rules working together: the Product to a Power Property squares the coefficient and variable separately, and the Power Property multiplies the exponents ( $u^3$ raised to the 2nd power gives $u^{3 \cdot 2} = u^6$ ). The resulting trinomial has degree 6 rather than the usual degree 2.

Updated 2026-04-21

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OpenStax Elementary Algebra

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