Answer:
see explanation
Step-by-step explanation:
Expand both factors and collect like term
Using Pascal' triangle with n = 6 to obtain the coefficients
1 6 15 20 15 6 1
Decreasing powers of 1 from [tex]1^{6}[/tex] to [tex]1^{0}[/tex]
Increasing powers of 3x from [tex](3x)^{0}[/tex] to [tex](3x)^{6}[/tex]
[tex]1+3x)^{6}[/tex]
= 1.[tex]1^{6}[/tex][tex](3x)^{0}[/tex] + 6.[tex]1^{5}[/tex][tex](3x)^{1}[/tex] + 15.[tex]1^{4}[/tex][tex](3x)^{2}[/tex] + 20.[tex]1^{3}[/tex][tex](3x)^{3}[/tex] + 15.1²[tex](3x)^{4}[/tex] + 6.[tex]1^{1}[/tex][tex](3x)^{5}[/tex] + 1.[tex]1^{0}[/tex][tex](3x)^{6}[/tex]
= 1 + 18x + 135x² + 540x³ + 1215[tex]x^{4}[/tex] + 1458[tex]x^{5}[/tex] + 729[tex]x^{6}[/tex]
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[tex](1-3x)^{6}[/tex]
= 1.[tex]1^{6}[/tex][tex](-3x)^{0}[/tex] + 6.[tex]1^{5}[/tex][tex](-3x)^{1}[/tex] + 15.[tex]1^{4}[/tex][tex](-3x)^{2}[/tex] + 20.[tex]1^{3}[/tex][tex](-3x)^{3}[/tex] + 15.1²[tex](-3x)^{4}[/tex] + 6.[tex]1^{1}[/tex][tex](-3x)^{5}[/tex] + 1.[tex]1^{0}[/tex][tex](-3x)^{6}[/tex]
= 1 - 18x + 135x² - 540x³ + 1215[tex]x^{4}[/tex] - 1458[tex]x^{5}[/tex] + 729[tex]x^{6}[/tex]
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Collecting like terms from both expressions
[tex](1+3x)^{6}[/tex] + [tex](1-3x)^{6}[/tex]
= 2 + 270x² + 2430[tex]x^{4}[/tex] + 1458[tex]x^{6}[/tex]
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(2)
Using Pascal's triangle with n = 5
1 5 10 10 5 1
Decreasing powers of 1 from [tex]1^{5}[/tex] to [tex]1^{0}[/tex]
Increasing powers of 2x from [tex](2x)^{0}[/tex] to [tex](2x)^{5}[/tex]
[tex](1+2x)^{5}[/tex]
= 1.[tex]1^{5}[/tex][tex](2x)^{0}[/tex] + 5.[tex]1^{4}[/tex][tex](2x)^{1}[/tex] + 10.[tex]1^{3}[/tex][tex](2x)^{2}[/tex] + 10.[tex]1^{2}[/tex][tex](2x)^{3}[/tex] + 5.[tex]1^{1}[/tex][tex](2x)^{4}[/tex]+ 1.[tex]1^{0}[/tex][tex](2x)^{5}[/tex]
= 1 + 10x + 40x² + 80x³ + 80[tex]x^{4}[/tex] + 32[tex]x^{5}[/tex]
