Multiplying a trinomial by a trinomial — lesson. Mathematics CBSE, Class 7.

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To multiply a trinomial by a trinomial, you need to multiply each term of one trinomial by each terms of another trinomial and add the resulting products.

\begin{array}{l} (a + b + c) \cdot (d + e + f) = \\ = a \cdot d + a \cdot e + a \cdot f + b \cdot d + b \cdot e + b \cdot f + c \cdot d + c \cdot e + c \cdot f \\ = ad + ae + af + bd + be + bf + cd + ce + cf \end{array}

When degrees are multiplied with the same bases, their indicators add up.

a^{m} \cdot a^{n} = a^{m + n}

Example:

\begin{array}{l} (7 + b + c) \cdot (4 a^{5} + 2 ab + c^{2}) \\ = 7 \cdot 4 a^{5} + 7 \cdot 2 ab + 7 c^{2} + b \cdot 4 a^{5} + b \cdot 2 ab + b \cdot c^{2} + c \cdot 4 a^{5} + c \cdot 2 ab + c \cdot c^{2} \\ = 28 a^{5} + 14 ab + 7 c^{2} + 4 a^{5} b + 2 a b^{2} + b c^{2} + 4 {ca}^{5} + 2 abc + c^{3} \end{array}

\begin{array}{l} (5 xy - 3 x^{2} + z) \cdot (x^{2} + y + 1) = \\ = 5 xy \cdot x^{2} + 5 xy \cdot y + 5 xy \cdot 1 - 3 x^{2} \cdot x^{2} - 3 x^{2} \cdot y - 3 x^{2} \cdot 1 + z \cdot x^{2} + z \cdot y + z \cdot 1 \\ = 5 x^{1 + 2} y + 5 x y^{1 + 1} + 5 xy - 3 x^{2 + 2} - 3 x^{2} y - 3 x^{2} + z x^{2} + zy + z \\ = 5 x^{3} y + 5 x y^{2} + 5 xy - 3 x^{4} - 3 x^{2} y - 3 x^{2} + z x^{2} + zy + z \end{array}

Pay attention to the signs!

\begin{array}{l} (5 x^{2} y - x + t^{2}) \cdot (2 x^{2} - y + t) = \\ = 5 x^{2} y \cdot 2 x^{2} + 5 x^{2} y \cdot (- y) + 5 x^{2} y \cdot t - x \cdot 2 x^{2} - x \cdot (- y) - x \cdot t + t^{2} \cdot 2 x^{2} + t^{2} \cdot (- y) + t^{2} \cdot t \\ = 5 \cdot 2 x^{2 + 2} y - 5 x^{2} y^{1 + 1} + 5 x^{2} yt - 2 x^{2 + 1} + xy - xt + 2 x^{2} t^{2} - t^{2} y + t^{2 + 1} \\ = 10 x^{4} y - 5 x^{2} y^{2} + 5 x^{2} yt - 2 x^{3} + xyxt + 2 x^{2} t^{2} - t^{2} y + t^{3} \end{array}

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2. Multiplying a trinomial by a trinomial

Theory: